EXAMPLE 1 Identify congruent parts

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Presentation transcript:

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

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. FG = QR 12 = 2x – 4 16 = 2x 8 = x

Use properties of congruent figures EXAMPLE 2 Use properties of congruent figures You know that ∠ F Q. m F = m Q 68 o = (6y + x) 68 = 6y + 8 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 .

EXAMPLE 3 Show that figures are congruent Then, 1 4 and 2 3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent. The diagram shows AJ CK , KD JB , and DA BC . By the Reflexive Property, JK KJ . All corresponding parts are congruent, so AJKD CKJB.

GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, 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°

GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH CDEF. 3. Show that PTS RTQ. SOLUTION In the given diagram PS QR, PT TR, ST TQ and Similarly all angles are to each other, therefore all of the corresponding points of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem.

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°

GUIDED PRACTICE for Examples 4 and 5 By the definition of congruence, what additional information is needed to know that NDC NSR. SOLUTION CN NR, CDN NSR, DCN NRS Given : (Proved from above sum) NDC NSR. Proved :

GUIDED PRACTICE for Examples 4 and 5 STATEMENT REASON CDN NSR DCN NRS Given DCN NRS Given Therefore DC RS, DN SN as angles are congruent their sides are congruent.