Whenever two figures have the same size and shape, they are called congruent. Triangles ABC and DEF are congruent. You can match up vertices like

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1 Unit 1: orresponding Parts in a ongruence Section 1: ongruent Figures Whenever two figures have the same size and shape, they are called congruent. F D E Triangles and DEF are congruent. You can match up vertices like!d!e!f This means that: orresponding ngles orresponding Sides! segment!! segment!! segment! orresponding parts of a congruent triangles are congruent. (PT) 1 Unit 1: orresponding Parts in a ongruence Section 1: ongruent Figures Example 1 Two triangles are congruent. D O a. ON b. N c. N d. segment ON e. segment N Example 2 Two triangles are congruent. D E M L N 43 4cm P O a. corresponds to b. m DN c. DN d. segment EN e. E= 2

2 Unit 1: orresponding Parts in a ongruence Section 2: Some Ways to Prove Triangles ongruent Postulate 12 (SSS Postulate) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Postulate 13 (SS Postulate) If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent. Postulate 14 (S Postulate) If two angles and an included side of one triangle are congruent to two angles and an included side of another triangle, then the triangles are congruent. 3 Unit 1: orresponding Parts in a ongruence Section 2: Some Ways to Prove Triangles ongruent Example 1 Which of the three postulates do you use? a. d. b. e. c. 4

3 Unit 1: orresponding Parts in a ongruence Section 3: Using ongruent Triangles Example 1 Given: segment and segment D bisect each other at M segment D segment 5 Unit 1: orresponding Parts in a ongruence Section 3: Using ongruent Triangles P Example 2 Given: segment PO plane ; segment PO bisects P segment D segment O 6

4 Unit 1: orresponding Parts in a ongruence Section 3: Using ongruent Triangles L Example 3 Given: m 1=m 2; m 3=m 4 M is the midpoint of segment JK J M K 7 Unit 2: Some Theorems ased on ongruent Triangles Section 4: The Isosceles Triangle Theorem 4-1 (The Isosceles Triangle Thm) If two sides of a triangles are congruent, then the angles opposite those sides are congruent. Proof Given: segment N segment N D 8

5 Unit 2: Some Theorems ased on ongruent Triangles Section 4: The Isosceles Triangle Theorem 4-1 (The Isosceles Triangle Thm) If two sides of a triangles are congruent, then the angles opposite those sides are congruent. vertex orollary 1 n equilateral triangle is also equiangular. vertex angles orollary 2 n equilateral triangle has three 60 angles. orollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. leg base angles base leg 9 Unit 2: Some Theorems ased on ongruent Triangles Section 4: The Isosceles Triangle Theorem 4-2 (The onverse of Isosceles Triangle Thm) If two angles of a triangles are congruent, then the sides opposite those angles are congruent. orollary 4 n equiangular triangle is also equilateral. How would you prove this? 10

6 Unit 2: Some Theorems ased on ongruent Triangles Section 4: The Isosceles Triangle Guided Practice Find the value of x. 2x-4 (1) (2) (3) 2x+2 30 x x x (4) Place the statements in an appropriate order for a proof. Given: 1N 2 segment OK N segment OJ (a) 3N 4 (b) 2N 4; 3N 1 (c) segment OK N segment OJ (d) 1N 2 O J 3 4 K Unit 2: Some Theorems ased on ongruent Triangles Section 4: The Isosceles Triangle Guided Practice (5) Do the proof of the following Given: segment Z N segment Y; ray YO bisects YZ; ray ZO bisects ZY segment YO N segment ZO Y 1 2 O 4 3 Z 12

7 Unit 2: Some Theorems ased on ongruent Triangles Section 5: Other Methods of Proving Triangles ongruent Theorem 4-3 (S Thm) If two sides and a non-included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Given: and DEF; N E; N F; segment N segment DF N DEF 13 Unit 2: Some Theorems ased on ongruent Triangles Section 5: Other Methods of Proving Triangles ongruent Theorem (HL Thm) If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. Given: and DEF; DE; EF and F are right s N DEF 14

8 Unit 2: Some Theorems ased on ongruent Triangles Section 6: Using More than One Pair of ongruent Triangle Example Given: 1N 2; 3N 4 segment TU N segment TW T U Q V W 15 Unit 2: Some Theorems ased on ongruent Triangles Section 6: Using More than One Pair of ongruent Triangle Example Given: segment RT N segment RV segment NT N segment NV segment TS N segment VS R 1 2 T N S V 16

9 Unit 2: Some Theorems ased on ongruent Triangles Section 6: Using More than One Pair of ongruent Triangle Example Given: segment RT N segment RV segment NT N segment NV segment TS N segment VS Prove by Paragraph Proof Given that segment RT N segment RV, segment NT N segment NV, and segment RN N segment RN by reflexive property, RNTN RNV by SSS. We note that segment RS N segment RS by reflexive property and 1N 2 by PT, so RTSN RVS by SS. Therefore, segment TS N segment VS by PT. R 1 2 T V N S 17 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors median of a triangle is a segment from a vertex of a triangle to the midpoint of the opposite side. n altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side. 18

10 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Draw three altitude on the following right triangle and obtuse triangle. perpendicular bisector of a segment is a line (or ray or segment) that is perpendicular to the segment at its midpoint. 19 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. m Theorem 4-6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment

11 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Given: line m is the bisector of segment ; is on m N m 21 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Given: = is on the bisector of segment

12 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line (or plane). R S t The length of segment RS, denoted RS, is the distance between the point P and the line t. 23 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-7 If a point lies on the perpendicular bisector of an angle, then the point is equidistant from the sides of the angle. P Z Y Theorem 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. P Y 24

13 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-7 If a point lies on the perpendicular bisector of an angle, then the point is equidistant from the sides of the angle. Given: ray Z bisects ; P lies on ray Z; P Z segment P ray ; segment PY ray Y P=PY 25 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Theorem 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. Given: segment P ray ; P segment PY ray ; P=PY ray P bisects Y 26

14 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors halkboard Examples Fill in the blank with always, sometimes, or never. 1. n altitude is perpendicular to the opposite side. 2. median is perpendicular to the opposite side. 3. n altitude is an angle bisector. 4. n angle bisector is perpendicular to the opposite side. 5. perpendicular bisector of a segment is equidistant form the endpoints of the segment. 6. Suppose ray OG bisects angle TOY. What can you deduce if you also know that: a. Point J lies on ray OG? b. point K is such that the distance from K to ray OT is 13 cm and the distance from K to ray OY is 13 cm? 27 Unit 2: Some Theorems ased on ongruent Triangles Section 7: Medians, ltitudes, and Perpendicular isectors Examples Given: ray DP bisects DE; ray EP bisects DE; ray P bisects You do!! D P E 28