Mathematics 10 Quarter 2 Module 5: Theorems on Secants, Tangents, Segments, and Sectors of a Circle

Mathematics Quarter 2 – Module 5: Theorems on Secants, Tangents, Segments, and Sectors of a Circle

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[illustration: students learning at home under a tree, with a teacher, a student reading beside a lamp, and a globe]

CO_Q2_Mathematics 10_ Module 5

Mathematics – Grade 10 Alternative Delivery Mode Quarter 2 – Module 5: Theorems on Secants, Tangents, Segments, and Sectors of a Circle First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Development Team of the Module

Writer’s Name: John Denver B. Pinkihan Co-Writer’s Name: Heather G. Banagui Reviewer’s Name: Bryan A. Hidalgo Management Team: May B. Eclar Benedicta B. Gamatero Carmel F. Meris Ethielyn E. Taqued Edgar H. Madlaing Marciana M. Aydinan Lydia I. Belingon

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Mathematics

Quarter 2 – Module 5: Theorems on Secants, Tangents, Segments, and Sectors of a Circle

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

[illustration: hand pointing at a circle, icon beside “What I Need to Know”]

What I Need to Know

This module contains activities, discussions and practice exercises on secants, tangents, segments and sectors of a circle.

After going through this module, you are expected to:

• Illustrate secants, tangents, segments, and sectors of a circle,
• Identify secant, tangents, segments, and sectors given an illustration,
• Proves theorems on secants, tangents and segments, and
• Find an unknown measurement of an angle and segments formed when secants and tangents intersect in a point inside, on and outside a circle.

What I Know

Direction: Read each item carefully and write the letter of the correct answer on a separate sheet of paper.

1. Which of the following line segments intersects a circle at exactly two points?
   A. line B. ray C. secant D. tangent

   
   

2. Which of the following is the region bounded by two radii of the circle and their intercepted arc?
   A. area B. circumference C. sector D. semicircle

   
   

3. How many pairs of common tangents can be drawn from the circles in Figure 1?

   
   

[illustration: two circles of different sizes, each with a center dot, side by side]

Figure 1

A. 1 B. 2 C. 3 D. 4

For items 4-8. In Figure 2, $\overline{AB}$, $\overline{AD}$, and $\overline{FG}$ are tangent segments of ⊙C

[chart: geometry diagram; Figure 2] Labels printed in the figure: A, B, C, D, E, F, G. Point C is the center of the circle. Segments AB, AD, and FG are tangent segments. Point A is outside the circle, and F is outside the circle.

Figure 2

4. Which of the following are the intercepted arcs of ∠DFG?
   A. $\overset{\frown}{DG}$ and $\overset{\frown}{DBG}$ C. $\overset{\frown}{BG}$ and $\overset{\frown}{EB}$
   B. $\overset{\frown}{EB}$ and $\overset{\frown}{DG}$ D. $\overset{\frown}{DG}$ and $\overset{\frown}{BG}$

   
   

5. If $m∠DAB=84$, then what is the $m∠DAC=$_______.
   A. 16° B. 31° C. 42° D. 168°

   
   

6. In the given figure, if measure of $\overline{AB}$ is 12. Which of the following is the measurement of $\overline{AD}$?
   A. 3 B. 6 C. 8 D. 12

   
   

7. Find the measurement of ∠F given that $\overset{\frown}{mDG}$ =78.
   a. 39° b. 102° c. 141° d. 156°

   
   

8. Which of the following is the $m∠ABC$?
   a. 100° b. 90° c. 60° d. 45°

   
   

For items 9-13. Refer to Figure 3.

[chart: geometry diagram; Figure 3] Labels printed in the figure: E, A, D, F, G, C, B. Point D is the center of the circle. A line through E and A passes through F and D, and a tangent line passes through B and C.

Figure 3

9. If $\overline{BC}$ = 12 and $\overline{GC}$ = 6, then which of the following is the length of $\overline{EG}$?
   a. 12 b. 18 c. 24 d. 72

   
   

10. If $\overline{AF}$ = 8, $\overline{FB}$ = 6, and $\overline{FG}$= 2, then which of the following is $\overline{EF}$?
    a. 10 b. 12 c. 16 d. 24

    
    

11. If $m∠AFG$ = 70 and $\overset{\frown}{mAG}$ = 80, then which of the following is $\overset{\frown}{mBE}$?
    a. 60° b. 70° c. 80° d. 220°

    
    

12. If $\overset{\frown}{mAB}$ = 110 , $\overset{\frown}{mAG}$ = 60 , and $\overset{\frown}{mBE}$ = 100, then $m∠C$=____.
    a. 100° b. 75° c. 50° d. 25°

    
    

13. If $\overset{\frown}{mAB}$ = 110, then which of the following is the $m∠ABC$?
    a. 55° b. 90° c. 110° d. 220°

    
    

14. The following statement is a theorem on two tangents intersecting outside a circle.

    
    

If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

[illustration: circle with center C; a tangent line touches the circle at B and passes through A; a secant from A passes through E and touches the circle at E; arc D marked on the lower left]

Which of the following is needed to show to prove the theorem? a. ∠BAE is an angle formed by $\overleftrightarrow{AB}$ and $\overleftrightarrow{AE}$ outside the circle. b. ∠BAE intercepts $\overset{\frown}{BE}$ and $\overset{\frown}{EDB}$. c. $\overset{\frown}{mBE}$ < $\overset{\frown}{mEDB}$ d. ∠BAE = $\frac{1}{2}$ ($\overset{\frown}{mEDB}$ – $\overset{\frown}{mBE}$)

15. Which of the following describes a major sector of a circle?
    a. A region bounded by two radii and an arc of the circle.
    b. A region bounded by two radii and an arc which measures 90°
    c. A region bounded by two radii and an arc which measures 180°
    d. A region bounded by two radii and an arc which measure 200°.

[illustration: hand holding a puzzle piece, icon beside “What’s In”]

What’s In

Before we proceed to our lesson, we need to review related concepts discussed in your Math 8 and concepts discussed in the previous modules. Activity 1 is a review on the Properties of Equality and Congruence, and Activity 2 is a review on chords, arcs, central angles, and inscribed angles.

Activity 1: Match Me

Match the mathematical statement in Column A by drawing an arrow to the corresponding property of equality or congruence in Column B. You can match more than one mathematical statement to one property.

Mathematical Statements	Property of Equality or Congruence
1. $\overline{AB}$ = $\overline{AB}$	a. Addition Property of Equality
2. $\overline{AB}$≅$\overline{AB}$	b. Subtraction Property of Equality
3. If $\overline{AB}$ ≅ $\overline{CD}$, then $\overline{AB}$ + $\overline{EF}$ ≅ $\overline{CD}$ + $\overline{EF}$	c. Transitive Property
4. If $\overline{AB}$ ≅ $\overline{CD}$, then $\overline{AB}$ – $\overline{EF}$ ≅ $\overline{CD}$ – $\overline{EF}$	d. Substitution Property
5. If ∠A ≅ ∠B and ∠B ≅ ∠C then ∠A≅∠C	e. Symmetric Property
6. If m∠A = m∠B and m∠B = m∠C then m∠A=m∠C	f. Reflexive Property
7. If $m\overline{AB}$ = $m\overline{DC}$ + $m\overline{EF}$ and $m\overline{EF}$= $m\overline{MN}$ then $m\overline{AB}$ = $m\overline{DC}$ + $m\overline{MN}$

Activity 2: Can you illustrate me?

In each item, draw a circle illustrating the following parts. Label and thicken the indicated part. You can use a colored pencil, crayon or marker.

1. Central angle A
2. Inscribed angle B
3. Points C and D on a circle
4. Points E and F outside a circle
5. Intercepted arc by a central angle G
6. Intercepted arc by an inscribed angle H

CO_Q2_Mathematics 10_ Module 5

What’s New

Activity 3: Touch or Divide?

Do you think tangent lines and secant lines are the same? To answer this question, consider the following illustrations of tangent lines and secant lines.

[illustration: four circles with lines; the upper-left circle is labeled A and touched by a tangent line, the upper-right circle is labeled B and touched by a vertical tangent line, the lower-left circle is labeled C and crossed by a secant line, and the lower-right circle is labeled D and crossed by a secant line]

In the figures, ʘA and ʘB are intersected by tangent lines while ʘC and ʘD are intersected by secant lines. Do you think there are similarities on tangent and secant lines? How about their differences?

Tangent Line A tangent to a circle is a line coplanar with the circle and intersects it in one and only one point. The point of intersection of the line and the circle is called the point of tangency.

[illustration: circle with center C; a line through points P and Q touches the circle at point A]

Figure 4

Example: In Figure 4, 𝑃𝑄⃡ intersects ⊙C at A only. Based on the definition of tangent line, 𝑃𝑄⃡ is a tangent line and A is the point of tangency.

Secant A secant is a line or segment or ray that intersects a circle at exactly two points. A secant contains chord of a circle.

[illustration: circle with center A; a line through points M and N crosses the circle at those two points]

Figure 5

Example: In Figure 5, 𝑀𝑁⃡ is a secant line of ʘA. It intersects the circle at two points.

What Is It

The following is the summary of what to be discussed in this section:

1. Definition of Terms
2. Postulate on Tangent Line
3. Theorems on Tangent Line
4. Theorems on Angles formed by Tangent Lines and Secant Lines
5. Theorems on Segments formed by Tangent Segments and Secant Segments

Common Tangent A common tangent is a line or segment or ray that is tangent to two circles in the same plane.

a. Common internal tangents intersect the segment joining the centers of the two circles.

[illustration: two circles labeled D and E with their centers joined by a dashed segment; two lines labeled c and d cross between the circles and are tangent to both]

Figure 6

Example: In Figure 6, 𝑐 and 𝑑 are both common internal tangents of ʘD and ʘE. Notice that they intersect the line connecting the center of ʘD and ʘE.

b. Common external tangents are lines or rays or segment that do not intersect the segment joining the centers of the two circles.

[illustration: two circles labeled M and N with their centers joined by a dashed segment; lines labeled t and s are tangent to both circles on the same side]

Figure 7

Example: In Figure 7, 𝑡 and 𝑠 are common external tangents. Note that they do not intersect the line connecting the centers of ʘM and ʘN.

External Secant Segment An external secant segment is the part of a secant segment that is outside of a circle.

[illustration: left, circle labeled C with a secant line h through point B and point A; right, circle labeled D with secant lines f and k, points E and F on the circle and point C outside the circle]

Figure 8

Example: In Figure 8, secant segments are illustrated outside ʘC and ʘD. In ʘC. 𝐴𝐵̅̅̅̅ is a segment of secant line ℎ and it is outside the circle. In ʘD, 𝐸𝐶̅̅̅̅ and 𝐹𝐶̅̅̅̅ are segments of secant line 𝑓 and secant line 𝑘 respectively outside the circle. For these reasons, 𝐴𝐵̅̅̅̅, 𝐸𝐶̅̅̅̅ and 𝐹𝐶̅̅̅̅ are external secant segments.

Activity 4: Tangent or Secant

To check your understanding on tangent and secant lines, supply what is being asked in the problems that follows.

1. In each figure, identify if the figure illustrates a tangent line, secant line, or neither.

[illustration: circle with a line touching it at one point, labeled a.] [illustration: circle with a nearly vertical line touching its right side, labeled d.]

b. [illustration: circle with a line crossing it, labeled b.]

c. [illustration: circle with a nearly vertical line passing through its center, labeled c.]

e. [illustration: circle with a line passing below and to the side of it without touching, labeled e.]

2. In the circle, name which lines are tangent and which lines are secant.

[illustration: circle with points S, N, T, A, E, L, J; a line through S and A, a line through N and L, a line through A and T, a line through J and L-side tangent]

3. In each figure, identify if common internal tangents, common external tangents or neither are being illustrated.

a. [illustration: two circles of different sizes with two lines tangent to both on the same side]

c. [illustration: a small circle above a large circle with two crossing lines tangent to both]

b. [illustration: a small circle above a large circle with two lines tangent to both, forming a widening shape]

d. [illustration: a small circle above a large circle with one line tangent to both on the left and one line tangent to the small circle and the large circle on the right]

4. In ʘC, identify which segments are external secant segment.

[illustration: circle with center C; points A, F, G, B, D, J, E, H; segments A-F-J-E, A-G-B, and D-H-B drawn with secant and tangent segments]

In item number 1, lines in letters a and d are tangent lines since they intersect or touch a circle at only one point

a. [illustration: circle with a line touching it at one point]

d. [illustration: circle with a nearly vertical line touching its right side]

Lines in letters b and c are secant lines since each intersects a circle at two points.

b. [illustration: circle with a line crossing it]

c. [illustration: circle with a nearly vertical line passing through its center]

Line in letter e is neither a tangent line nor a secant line since it does not touch any point on a circle.

e. [illustration: circle with a line passing below and to the side of it without touching]

In item number 2, the only tangent line is 𝐽𝐿⃡ while the secant lines are 𝑆𝐽⃡ , 𝑁𝐿⃡ and 𝐴𝑇⃡ .

[illustration: circle with points S, N, T, A, E, L, J; lines through S and A, N and L, A and T, and a line through J tangent at L]

In item number 3, letter d illustrates lines that are common internal tangents,

c. [illustration: a small circle above a large circle with two crossing lines tangent to both]

letter b illustrates lines that are common external tangents,

b. [illustration: a small circle above a large circle with two lines tangent to both, forming a widening shape]

and letter a and letter e both neither illustrate common internal tangents nor common external tangents.

a. [illustration: a small circle and a large circle with two lines, one tangent to both circles on one side and one tangent to only the large circle and the small circle]

d. [illustration: a small circle above a large circle with one line tangent to both on the left and one line tangent to the small circle and the large circle on the right]

In item 4, 𝐴𝐸̅̅̅̅ is only the external secant line. 𝐀𝐅̅̅̅̅, and 𝐉𝐄̅̅̅ are segments outside ʘC. Thus, the external segments of a secant line are 𝐀𝐅̅̅̅̅, and 𝐉𝐄̅̅̅.

Postulate on Tangent Line

At a given point on a circle, one and only one line can be drawn that is tangent to the circle.

[illustration: circle with center R; a line through points A and B touches the circle at point C]

Figure 9

To illustrate, consider ⊙R in Figure 9. If C is a point on the circle, then 𝐴𝐵̅̅̅̅ is the only line that can be drawn through C that is tangent to ʘR.

Theorems on Tangent Line

1. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

[illustration: circle with center R; a line through points A and B touches the circle at point C, and segment CR is drawn to the center]

Figure 10

In Figure 10, 𝐴𝐵̅̅̅̅ is tangent to ʘR, at C. If we connect C to R, then 𝐶𝑅̅̅̅̅⊥𝐴𝐵̅̅̅̅

2. If a line is perpendicular to the radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

Figure 11

[illustration: circle Q with a line through point E on the circle; the line runs through points C and D, and a radius QE is drawn]

ʘQ in Figure 11, DC̅̅̅̅⊥QE̅̅̅̅and QE̅̅̅̅ is a radius of circle Q. Therefore, DC̅̅̅̅ is tangent to ʘQ.

3. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.

[illustration: circle A with external point C; tangent segments from C touch the circle at D and B]

Figure 12

In Figure 12, DC̅̅̅̅ and BC̅̅̅̅ are tangent to circle A at D and B, respectively, from the common external point, C. From the theorem, BC̅̅̅̅ ≅ DC̅̅̅̅.

Theorems on Angles Formed by Tangents and Secants

1. If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

[illustration: circle C with two tangent lines BA and AE intersecting outside the circle at point A; the tangent points are B and E, and point D lies on the far arc]

Figure 13

In Figure 13, tangent BA⃡ and tangent AE⃡ intersect outside ʘC at point A. BE⌒ and EDB⌒ are the two intercepted arcs of ∠BAE. The theorem states that:

m ∠BAE = 1/2 (m EDB⌒ − m BE⌒)

2. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

[illustration: circle with two secants intersecting outside the circle at point P; the points on the circle are N, O, X, Y, and M]

Figure 14

In Figure 14, NX⃡ and MY⃡ are two secants intersecting outside the circle at point P. XY⌒ and MN⌒ are the two intercepted arcs of ∠XPY. The theorem states that:

m ∠XPY = 1/2 (m XY⌒ − m MN⌒)

3. If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the difference of the measures of the intercepted arcs.

[illustration: circle P with a secant through points N and Q and a tangent through point L and point O; point M lies on the lower arc]

Figure 15

In Figure 15, secant NL⃡ and tangent LO⃡ intersect outside the circle at point L. OMN⌒ and QO⌒ are the two intercepted arcs of ∠NLO. The theorem states that:

m ∠NLO = 1/2 (m OMN⌒ − m QO⌒)

4. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

[illustration: circle with two secants WS and RX intersecting inside the circle; points W, R, X, S are on the circle, and the angles are labeled 1 and 2]

Figure 16

In Figure 16, WS⃡ and RX⃡ are two secants intersecting inside the circle. WR⌒ and XS⌒ are the two intercepted arcs of ∠1 while XW⌒ and RS⌒ are the two intercepted arcs of ∠2. The theorem states that:

m ∠1 = 1/2 (m WR⌒ + m XS⌒) m ∠2 = 1/2 (m XW⌒ + m RS⌒)

5. If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc.

[illustration: circle with tangent line ZP touching the circle at Y, a secant through Y and Q, and points X and P on the circle and line]

Figure 17

In Figure 17, ZP⃡ is tangent at the circle at Y. Secant QY⃡ and tangent ZP⃡ intersect at Y. QY⌒ is the intercepted arc of ∠ZYQ while YXQ⌒ is the intercepted arc of ∠QYP. The theorem states that:

m ∠ZYQ = 1/2 m QY⌒ m ∠QYP = 1/2 m YXQ⌒

Theorems on Secant Segments, Tangents Segments, and External Segments

1. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.

[illustration: circle A with two secant segments drawn from external point C; the points on the circle are B, E, D, and F]

Figure 18

In ʘA of Figure 18, BC̅̅̅̅ and DC̅̅̅̅ are secant segments from external point C. EC̅̅̅̅ is the external secant segment of BC̅̅̅̅ and FC̅̅̅̅ is the external secant segment of DC̅̅̅̅. The theorem states that:

(BC̅̅̅̅)•(EC̅̅̅̅) = (DC̅̅̅̅)•(FC̅̅̅̅)

2. If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.

[illustration: circle A with a tangent segment EC and a secant segment DC through point G, drawn from external point C]

Figure 19

In ʘA of Figure 19, EC̅̅̅̅ and DC̅̅̅̅ are tangent segment and secant segment respectively from external point C. GC̅̅̅̅ is the external secant segment of DC̅̅̅̅. The theorem states that:

(EC)² = (DC̅̅̅̅)•(GC̅̅̅̅)

3. If two secant lines intersect inside the circle, the product of the lengths of segments formed inside the circle are equal.

[illustration: circle with secant lines GA and CR intersecting inside the circle at point E; points G, R, A, and C are on the circle]

Figure 20

In Figure 20, secant lines GA⃡ and CR⃡ intersect inside the circle. GE̅̅̅, EA̅̅̅, RE̅̅̅̅ and EC̅̅̅̅ are the segments formed inside the circle. Thus,

(GE̅̅̅)• (EA̅̅̅) = (RE̅̅̅̅)• (EC̅̅̅̅)

Activity 5: Match Me

Match illustrations in column A with the relationship about its angles formed, tangent segment, and secant segments in column B. An illustration in column A can be matched to more than one relationship in column B. Or more than one illustration in column A can be match to one relationship in column B. In addition, any line segment that looks like a tangent line is tangent, any line segment that looks like secant line is secant and a point that looks like a point of tangency is really the point of tangency.

[transcriber’s note: Column A holds five numbered illustrations, which continue on the next page. The printed sheet places them beside the Column B statements, but the row pairing in the table below is not printed and does not show which statement matches which illustration.]

Column A	Column B
	a. (DC̅̅̅̅)² = (FC̅̅̅̅)(HC̅̅̅̅)

[transcriber’s note: table continued from the previous page]

Column A	Column B
1. [illustration: circle A with a tangent line BC touching the circle at D; radius AD is drawn]	b. (DC̅̅̅̅)(JC̅̅̅̅) = (FC̅̅̅̅)(HC̅̅̅̅)
2. [illustration: circle A with tangent segments BC and EC and points D, F, G, and E; secant lines through C]	c. DC̅̅̅̅ ≅ FC̅̅̅̅
3. [illustration: circle A with tangent segment BC touching at D, and secant segments through C, E, F, and H]	d. If BC̅̅̅̅ is tangent to ʘA at D, then BC̅̅̅̅⊥AD̅̅̅̅
4. [illustration: circle A with secant segments BC and EC and points D, J, H, F, and G]	e. In ʘA, if BC̅̅̅̅⊥AD̅̅̅̅, then BC̅̅̅̅ is tangent to the circle at D
5. [illustration: circle with a secant line through points D, B, F, and E, and a second line through B, F, and C]	f. m ∠BCE = 1/2 (m FGD⌒ − m DF⌒)
	g. m ∠BCE = 1/2 (m FGD⌒ − m JH⌒)
	h. m ∠BCE = 1/2 (m FGD⌒ − m DH⌒)
	i. (DF̅̅̅̅)• (FE̅̅̅̅) = (BF̅̅̅̅)• (FC̅̅̅̅)

Did you answer the Match Me activity correctly?

After illustrating the theorems, we consider the proof of some in the next discussions.

Proof

1. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

Step 1: To prove the theorem, it would be easier to transform to an equivalent if-then statement using Figure 21.

[illustration: circle O with two secants intersecting outside at point P; points N, M, X, and Y are marked, with dashed segments connecting N to Y and M to X]

Figure 21

Step 1: If-then: If NX⃡ and MY⃡ are two secants intersecting outside the circle at point P, XY⌒ and MN⌒ are the two intercepted arcs of ∠XPY then

m ∠XPY = 1/2 (m XY⌒ − m MN⌒)

Step 2: Identify the given and what to prove. In the if-then statement, the if statement is the given and the then statement is the conclusion.

Step 2: Given: NX⃡ and MY⃡ are two secants intersecting outside the circle at point P, XY⌒ and MN⌒ are the two intercepted arcs of ∠XPY

Need to Show: m ∠XPY = 1/2 (m XY⌒ − m MN⌒)

Step 3: Prove now the theorem

Statement	Reason
1. Connect N and Y, and M and X	By construction
2. ∠XMY is an exterior angle of ΔPXM.	Definition of an exterior angle of a triangle.
3. m∠XMY = m∠XPY + m∠NXM	The measure of the exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.
4. MN⌒ is intercepted by inscribed ∠NXM and XY⌒ is intercepted by inscribed ∠XMY	Definition of intercepted arc
5. m∠NXM = 1/2 m MN⌒ m∠XMY = 1/2 m XY⌒	The measure of an inscribed angle is one half the measure of its intercepted arc.
6. 1/2 m XY⌒ = m∠XPY + 1/2 m MN⌒	Substitution from statements 3 to 5
7. 1/2 m XY⌒ − 1/2 m MN⌒ = m∠XPY	Subtraction Property

[transcriber’s note: table continued from the previous page]

Statement	Reason
8. m∠XPY = 1/2 (m XY⌒ − m MN⌒)	By factoring

■

2. If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc.

Following the steps given in example 1,

[illustration: circle O with tangent line ZP touching the circle at Y, a secant through Y and Q, dashed radii OY and OQ, and points X and P marked]

Figure 22

Step 1: If,

• ZP⃡ is tangent to ʘO at Y, and
• Secant QY⃡ and tangent ZP⃡ intersect at Y then

m ∠ZYQ = 1/2 m QY⌒ and m ∠QYP = 1/2 m YXQ⌒

Step 2: Given: ZP⃡ is tangent to ʘO at Y Secant QY⃡ and tangent ZP⃡ intersect at Y

Need to Show: m ∠ZYQ = 1/2 m QY⌒ and m ∠QYP = 1/2 m YXQ⌒

Proof: Since we need to show two relationships, the proof of the theorem shall be divided into two. The first part of the proof is to show that m ∠ZYQ = 1/2 m QY⌒. Consider the proof that follows

Statement	Reasons
1. Connect O to Q and O to Y forming OQ̅̅̅̅ and OY̅̅̅̅	By construction
2. YQ⌒ is intercepted by ∠YOQ	Definition of intercepted arc.
3. m∠YOQ = m YQ⌒	The measure of a central angle is equal to the measure of its intercepted arc.
4. ZP⃡ is tangent to the circle at Y	Given
5. OY̅̅̅̅⊥ZP⃡	If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
6. m∠ZYO = 90	Perpendicular lines form a 90° angle.
7. ∠ZYQ + ∠QYO = ∠ZYO	Whole part postulate
8. m∠ZYQ = m∠ZYO − m∠QYO	Subtraction Property
9. m∠ZYQ = 90 − m∠QYO	Substitution from statements 6 to 8

[transcriber’s note: table continued from the previous page]

Statement	Reasons
10. m∠QYO = 90 − m∠ZYQ	Addition property of equality
11. YO̅̅̅̅ and OQ̅̅̅̅ are radii of ʘO	Definition of radius of a circle
12. YO̅̅̅̅ = OQ̅̅̅̅	Radii of a circle are equal
13. ΔYOQ is an isosceles triangle	At least two sides of an isosceles triangle are equal
14. m∠QYO = m∠OQY	Angles opposite the equal sides of an isosceles triangles are equal.
15. m∠QYO + m∠OQY + m∠YOQ = 180	The sum of the interior angles of a triangle is 180
16. m∠QYO + m∠QYO + m∠YOQ = 180	Substitution from statements 14 to 15
17. 2m∠QYO + m∠YOQ = 180	Simplification
18. 2(90 − m∠ZYQ) + m∠YOQ = 180	Substitution from statements 10 to 17
19. 180 − 2 m∠ZYQ + m∠YOQ = 180	Distributive Property
20. − 2 m∠ZYQ + m∠YOQ = 0	Subtraction Property
21. m∠YOQ = 2 m∠ZYQ	Addition Property
22. 1/2 m∠YOQ = m∠ZYQ	Division Property
23. m∠YOQ = m YQ⌒	A central angle is equal to its intercepted arc.
24. 1/2 m YQ⌒ = m∠ZYQ	Substitution from statements 23 to 22

The first part is done for you, prove the second part of the theorem.

Activity 6: Can You Complete by Proof?

The other theorems will be left for you to prove.

The last concept we need to discuss is about sectors of a circle.

Sector of a Circle

[illustration: three circles with center A and points B, C, and D on the circle; the first shows a central angle BAC, the second shades the minor sector bounded by radii AB and AC and arc BC, and the third shades the major sector]

Figure 23

The region of the circle bounded by radii and an arc are called sectors, as illustrated Figure 23. The shaded region in the center circle is a minor sector since it

is bounded by a minor arc and radii. While the shaded region in the right most figure is a major sector since it is bounded by two radii and a major arc.

What’s More

Activity 7. Apply the different theorems discussed to solve for the unknown in the following exercises.

1. If m⌒MKL = 220°, what is m∠MQL?

[illustration: circle with points K, L, M on it, and exterior point Q with a tangent line through M and a secant through L]

2. Suppose m⌒CG = 6x + 5, m⌒RA = 4x + 15, and m∠AEC = 120, Find
   a. x b. m⌒CG c. m⌒RA

[illustration: circle with points C, G, E, R, A; a horizontal line through C, E, R and a secant through G, E, A]

3. If m⌒QNO = 238°, what is m∠PQO? m∠PQR?

What I Have Learned

In the previous activities that you have done, were you able to apply the theorems you have learned? In the next activity you are to summarize now your understanding about the theorems.

I think I do!

Answer the following questions.

1. What is the relationship of two secants intersecting in the exterior of a circle to the measures of its intercepted arcs?

   
   

2. What is the relationship of a secant and a tangent intersecting in the exterior of a circle to its intercepted arcs?

   
   

3. What is the relationship of two secants intersecting in the interior of a circle to the measures of the intercepted arcs and its vertical angles?

   
   

4. What is the relationship among the segments formed inside a circle when two secant lines intersect in the interior of a circle?

   
   

5. How are the two secants segments drawn from the exterior point to the circle related to their external secant segments?

   
   

6. How about if a tangent segment and a secant segment are drawn to the circle from an exterior point, what will be its relationship to its segments formed?

   
   

[illustration: circle with points N, O, Q, P, R; a secant through O, Q, R and a tangent through Q and P]

What I Can Do

Did you know that the tire of a car touches the road at a point when running? Or in other words, a tire is tangent on the road. What do you think is the reason?

Can you cite 5 other situations where tangent line or secant line is applied around us?

Assessment

Directions: Read carefully each item and write the letter of the correct answer on a separates sheet of paper

1. Which of the following line segments intersects a circle at exactly two points?
   A. line B. ray C. secant D. tangent

   
   

2. Which of the following is the region bounded by two radii of the circle and their intercepted arc?
   A. area B. circumference C. semicircle D. sector

   
   

For items 3-7. Refer to the figure at the right.

AB, AD, and FG are tangent segments of ⊙C.

[illustration: circle C with tangent segments from exterior point A to points B and D, and a tangent through F and G; points E, F, G, D, B labeled]

3. Which of the following are the intercepted arcs of ∠AFG?
   A. D⌒G and D⌒BG C. B⌒G and E⌒B
   B. E⌒B and D⌒G D. D⌒G and B⌒G

   
   

4. If m∠DAB=84, then m∠DAC=_______.
   A. 16° B. 42° C. 51° D. 168°

   
   

5. In the figure given that AB is 12. Which of the following is the measurement of AD?
   A. 3 B. 6 C. 12 D. 24

   
   

6. Find the measurement of ∠F given m⌒DG = 78.
   A. 39° B. 102° C. 141° D. 156°

   
   

7. Which of the following is the m∠ABC?
   A. 100° B. 90° C. 60° D. 45°

   
   

For items 8-15. Refer to the figure at the right.

[illustration: circle with points E, A, D (center), F, G, C, B; a secant through E, D, F, A and a secant/tangent line through B, F, G, C]

8. If BC = 12 and GC = 6, then which of the following is the length of EG?
   A. 12 B. 18 C. 24 D. 72

   
   

9. If AF = 8, FB = 6, and FG = 2, then which of the following is EF?
   A. 10 B. 12 C. 16 D. 24

   
   

10. If m∠AFG = 70 and m⌒AG = 80, then which of the following is the m⌒BE?
    A. 60° B. 70° C. 80° D. 220°

    
    

11. If m⌒AB = 110, m⌒AG = 60, and m⌒BE = 100, then m∠C=____.
    A. 100° B. 75° C. 50° D. 25°

    
    

12. If m⌒AB = 110, then which of the following is the m∠ABC?_.
    A. 55° B. 90° C. 110° D. 220°

    
    

13. Which of the following arcs is an intercepted arc of ∠EFA?
    A. EA B. AGB C. BG D. BEA

    
    

14. In the figure which of the following represents a tangent?
    A. BC B. EC C. BA D. BF

    
    

15. In the figure which of the following represents a secant?
    A. BC B. EC C. BA D. BF

    
    

Additional Activity

Mathematical Reasoning

Investigate and explain what is wrong with the student’s solution below.

1. Solution: x/2 = 4/4
   4x = 4(2)
   x = 2

[illustration: circle with two intersecting secant lines; segment labels 2, 4, 4, and x]

2. Solution: x² = 4(10)
   x² = 40
   x = √40
   x = 2√10

[illustration: circle with a secant and an external secant from one point; segment labels 4, 10, and x]

Answer Key

[transcriber’s note: this page is printed upside down; the answer boxes are transcribed below in normal reading orientation]

What I Know

1. C
2. C
3. B
4. A
5. C
6. D
7. B
8. B
9. B
10. D
11. A
12. D
13. A
14. D
15. D

What’s In

Activity 1

1. f
2. f
3. a
4. b
5. c
6. c
7. d

What’s It

Activity 5

1. d,e
2. c,f
3. a,h
4. b,g
5. i

ASSESSMENT

1. C
2. D
3. A
4. B
5. C
6. B
7. B
8. B
9. D
10. A
11. D
12. A
13. A
14. A
15. B

What’s More

Activity 7

1. m∠MQL = ½(m⌒MKL − m⌒ML) = ½(220 − 140) = 40
2. ∠AEC and ∠CEG form a Linear Pair. It is given that m∠AEC is 120 then m∠CEG is 60. In addition, m∠CEG = ½(m⌒CG + m⌒RA). Substituting the given and m∠CEG, then x=10, CG = 65 and AR =55.
3. m∠PQO = 61°
   m∠PQR = 119°

ADDITIONAL ACTIVITY

1. From the theorem, when two secants intersect inside the circle, 2(x) = 4(4) thus x = 8
2. From the theorem, when two secants intersect outside the circle, x² = 4(16) thus x = 8

References

Nivera, Gladys C., Ph.D. and Minie Rose C. Lapinid, Ph.D.. 2015. Grade 10 Mathematics Patterns and Practicalities. SalesianaBooks. Makati City. Don Bosco Press, Inc.

Callanta, M.M. Et.Al. Mathematics- Garde 10 Learners Module, 2015, REX Bookstore, Inc. Pasig City. November 6, 2019.

Ulpina, Jisela N., and Razon, Larida-M E, 2015. Grade 10 Math Builders New Enhanced Edition. Mc Arthur Highway, Dalandan, Valenzuela City.Jo-Es Publishing House Inc.

Oronce, Orlando and Mendoza, Marilyn O. 2015. RBS Mathematics. E- Math 10. Nicanor Reyes St. Sampaloc, Manila. Rex Books Store, Inc

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