1. When are two angles congruent?

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Presentation transcript:

1. When are two angles congruent? ANSWER when they have the same measure 2. In ∆ABC, if m A = 64º and m B = 71º, what is m C? ANSWER 45º 3. What property of angle congruence is illustrated by this statement? If A B and B C, then A C. ANSWER Transitive Property

Proving triangles congruent. 4.2 Identify congruent figures. Target Proving triangles congruent. GOAL: 4.2 Identify congruent figures.

Vocabulary congruent figures - figures the same shape and size; two figures are congruent if and only if all corresponding angles and all corresponding sides are congruent. CPCTC - Corresponding Parts of Congruent Triangles are Congruent Third Angles Theorem 4.3 – If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. R-S-T Theorem for Congruent Triangles 4.4 – Congruence of triangles is reflexive, symmetric, and transitive.

EXAMPLE 1 Identify congruent parts To write a congruence statement for the triangles first identify all pairs of congruent corresponding parts. SOLUTION Corresponding angles J T, ∠ K S, L R Corresponding sides JK TS, KL SR, LJ RT congruence statement The diagram indicates that JKL TSR.

Use properties of congruent figures EXAMPLE 2 Use properties of congruent figures In the diagram, DEFG SPQR. Find the value of x. Find the value of y. SOLUTION You know that FG QR. You know that ∠ F Q. FG = QR m F = m Q 12 = 2x – 4 68 o = (6y + x) 16 = 2x 68 = 6y + 8 8 = x 10 = y

EXAMPLE 3 Show that figures are congruent PAINTING If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain. SOLUTION From the diagram, A C and D B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC .

GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the below, ABGH CDEF. Identify all pairs of congruent corresponding parts. SOLUTION Corresponding sides: AB CD, BG DE, GH FE, HA FC Corresponding angles: A C, B D, G E, H F.

GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH CDEF. 2. Find the value of x and find m H. SOLUTION (a) You know that H F (4x+ 5)° = 105° 4x = 100 x = 25 (b) You know that H F m H m F =105°

EXAMPLE 4 Use the Third Angles Theorem Find m BDC. SOLUTION A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, m ACD = 180° – 45° – 30° = 105° . So, m ACD = m BDC = 105° by the definition of congruent angles. ANSWER

EXAMPLE 5 Prove that triangles are congruent Write a proof. GIVEN AD CB, DC AB ACD CAB, CAD ACB PROVE ACD CAB Plan for Proof AC AC. Use the Reflexive Property to show that Use the Third Angles Theorem to show that B D

Prove that triangles are congruent EXAMPLE 5 Prove that triangles are congruent Plan in Action STATEMENTS REASONS AD CB , DC BA Given AC AC. Reflexive Property of Congruence ACD CAB, CAD ACB Given B D Third Angles Theorem ACD CAB Definition of

GUIDED PRACTICE for Examples 4 and 5 DCN. In the diagram, what is m SOLUTION CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75° By the definition of congruence, what additional information is needed to know that NDC NSR. ANSWER DC RS and DN SN