1. Other Angle Relationships in Circles
Geometry Section 8 Day 3

2. Theorem 10.11
If a tangent and a chord intersect at a point on the circle, then the measure of angle formed is one half the measure of its intercepted arc.
𝑚∠1= 1 2 𝑚 𝐴𝐵
𝑚∠2= 1 2 𝑚 𝐵𝐶𝐴
Geometry S8 Day 3

3. Theorem 10.12- Angles Inside the Circle Theorem
Geometry S8 Day 3
If two chords intersect inside a circle, the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle
𝑚∠1= 1 2 𝑚 𝐷𝐶 +𝑚 𝐴𝐵
𝑚∠2= 1 2 (𝑚 𝐵𝐶 +𝑚 𝐴𝐷 )

4. Theorem 10.13- Angles Outside the Circle Theorem
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs
𝑚∠1= 1 2 𝑚 𝐵𝐶 −𝑚 𝐴𝐶
𝑚∠2= 1 2 𝑚 𝑃𝑄𝑅 −𝑚 𝑃𝑅
𝑚∠3= 1 2 (𝑚 𝑋𝑌 −𝑚 𝑊𝑍 )
Geometry S8 Day 3

5. Example 1: Solve for the ? 68°= 1 2 80°+𝑚 𝑊𝑀 68°=40°+ 1 2 𝑚 𝑊𝑀
68°= °+𝑚 𝑊𝑀
68°=40°+ 1 2 𝑚 𝑊𝑀
68°−40°= 1 2 𝑚 𝑊𝑀
28°= 1 2 𝑚 𝑊𝑀
2∙28°=𝑚 𝑊𝑀
56°=𝑚 𝑊𝑀
?=56°
Geometry S8 Day 3

6. Example 2: Solve for x 40°= 1 2 124°− 5𝑥−6 2∙40°=124°− 5𝑥−6
40°= °− 5𝑥−6
2∙40°=124°− 5𝑥−6
80°=124°−5𝑥+6
80°−124°−6=−5𝑥
−50=−5𝑥
−50 −5 = −5𝑥 −5
𝑥=10
Geometry S8 Day 3

7. Homework
Geometry S8 Day 3
Assignment 8-3

8. Segment Lengths in Circles
Geometry Section 8 Day 3

9. Theorem 10.14- Segments of Chords Theorem
Geometry S8 Day 3
If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord
𝐸𝐴∙𝐸𝐵=𝐸𝐶∙𝐸𝐷

10. Theorem 10.15
If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
EA∙𝐸𝐵=𝐸𝐶∙𝐸𝐷
Geometry S8 Day 3

11. Theorem 10.16- Segments of Secants and Tangents Theorem
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.
𝐸 𝐴 2 =𝐸𝐶∙𝐸𝐷
Geometry S8 Day 3

12. Example 3: Solve for x 24∙14=16∙𝑥 336=16𝑥 336 16 = 16𝑥 16 𝑥=21
Geometry S8 Day 3

13. Example 4: Find the indicated segment
20 2 =16∙ 16+ 2𝑥−3
400=16∙ 16+2𝑥−3
400=16∙ 13+2𝑥
400=208+32𝑥
400−208=32𝑥
192=32𝑥
= 32𝑥 32
𝑥=6
𝐺𝐸= − =16+12−3=𝟐𝟓
Geometry S8 Day 3

14. Homework
Geometry S8 Day 3
Assignment 8-4