Splash Screen. Lesson Menu Five-Minute Check (over Lesson 10–3) Then/Now New Vocabulary Theorem...

Five-Minute Check (over Lesson 10–3)

Then/Now

New Vocabulary

Theorem 10.6: Inscribed Angle Theorem

Proof: Inscribed Angle Theorem (Case 1)

Example 1: Use Inscribed Angles to Find Measures

Theorem 10.7

Example 2: Use Inscribed Angles to Find Measures

Example 3: Use Inscribed Angles in Proofs

Theorem 10.8

Example 4: Find Angle Measures in Inscribed Triangles

Theorem 10.9

Example 5: Real-World Example: Find Angle Measures

Over Lesson 10–3

D. 22.5

Over Lesson 10–3

A. 24.6

B. 26.8

C. 28.4

D. 30.2

Over Lesson 10–3

You found measures of interior angles of polygons. (Lesson 6–1)

• Find measures of inscribed angles.

• Find measures of angles of inscribed polygons.

• inscribed angle

• intercepted arc

Use Inscribed Angles to Find Measures

A. Find mX.

Answer: mX = 43

= 2(52) or 104

D. 188

A. Find mC.

ALGEBRA Find mR.

R S R and S both intercept . mR mS Definition of congruent angles

12x – 13 = 9x + 2 Substitutionx = 5 Simplify.

Answer: So, mR = 12(5) – 13 or 47.

ALGEBRA Find mI.

Use Inscribed Angles in Proofs

Write a two-column proof.

Given:

Prove: ΔMNP ΔLOP

1. Given

Proof:Statements Reasons

LO MN2. If minor arcs are congruent, then corresponding chords

are congruent.

Use Inscribed Angles in Proofs

M L 4. Inscribed angles of the same arc are congruent.

MPN OPL 5. Vertical angles are congruent.

ΔMNP ΔLOP 6. AAS Congruence Theorem

3. Definition of intercepted arcM intercepts and

L intercepts .

Write a two-column proof.

Given:

Prove: ΔABE ΔDCE

Select the appropriate reason that goes in the blank to complete the proof below.

1. Given

AB DC 2. If minor arcs are congruent, then corresponding chords are congruent.

D A 4. Inscribed angles of the same arc are congruent.

DEC BEA 5. Vertical angles are congruent.

ΔDCE ΔABE 6. ____________________

3. Definition of intercepted arcD intercepts and

A intercepts .

A. SSS Congruence Theorem

B. AAS Congruence Theorem

C. Definition of congruent triangles

D. Definition of congruent arcs

Find Angle Measures in Inscribed Triangles

ALGEBRA Find mB.

ΔABC is a right triangle because C inscribes a semicircle.

mA + mB + mC = 180 Angle Sum Theorem(x + 4) + (8x – 4) + 90 = 180 Substitution

9x + 90 = 180 Simplify.9x = 90 Subtract 90 from each

side.x = 10 Divide each side by 9.

Answer: So, mB = 8(10) – 4 or 76.

ALGEBRA Find mD.

Find Angle Measures

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

Find Angle Measures

Since TSUV is inscribed in a circle, opposite angles are supplementary.

mS + mV = 180 mU + mT = 180 mS + 90 = 180 (14x) + (8x + 4) = 180

mS = 90 22x + 4 = 18022x = 176

x = 8Answer: So, mS = 90 and mT = 8(8) + 4 or 68.

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mN.

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