4-2 Triangle Congruence Conditions. Congruent Triangles - C F. and

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Lionel York
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1 4-2 Triangle ongruence onditions ongruent Triangles -,, ª is congruent to ª (ª ª) under a correspondence of parts if and only if 1) all three pairs of corresponding angles are congruent, and 2) all three pairs of corresponding sides are congruent. and Three properties of congruence hold for congruent triangles: 1) Reflexive Property ª ª for all triangles ª. 2) Symmetric Property If ª ª, then ª ª. 3) Transitive Property If ª ª and ª ªGHI, then ª ªGHI. There is a special case with an equilateral triangle, ª, since all the sides and all the angles are congruent to each other. What are the six congruences? In order to draw a triangle we only need a partial triangle which we will complete. Given a partial triangle, you will find that only one triangle can be made from those measurements. Given two sides and the included angle, you only need to draw the third side to complete the triangle. There will be only one triangle that can be drawn! Postulate 4.1 SS ongruence Postulate If two sides and the included angle of one triangle are congruent to corresponding parts of another triangle, then the two triangles are congruent. Q Y P R 4 X Z 4

2 If the included angle is a right angle, then you only need the length of both legs to make congruent triangles. Theorem 4.2 LL ongruence Theorem If two legs of one right triangle are congruent to corresponding parts of another right triangle, then the two triangles are congruent. nd in the case where we know the length of the hypotenuse and one leg, we can find the length of the third leg using the Pythagorean Theorem and then use Theorem 4.2 Theorem 4.3 HL ongruence Theorem If the hypotenuse and one leg of one right triangle are congruent to corresponding parts of another right triangle, then the two triangles are congruent. Once we prove that two triangles are congruent, then we can assume that all pairs of corresponding parts are congruent. We indicate this using the abbreviation PT for orresponding Parts of ongruent Triangles are ongruent. xample: Given: and Prove:

3 xample: Given: JL LM, LJ JK, MJ KL Prove: JLM LJK M L J K 1. JL LM LJ JK 2. JLM and MLJ and KJL LJK are right angles 2. are right triangles MJ KL LJ LJ JLM LJK 6. xample: Given: m H = m GH = 90 and HG Prove: GH H G indicate what you know! H

4 efore we go on, here are a few more definitions: 4-2 Triangle ongruence onditions- Part 2 efinition of Perpendicular Two sets are perpendicular if (1) each of them is a line, a ray, or a segment, (2) they intersect, and (3) the lines containing them are perpendicular. Perpendicular lines are two lines that intersect to form right angles. M perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments. In the diagram above, is the perpendicular bisector of. n angle bisector is a ray that divides an angle into two congruent coplanar angles. Its endpoint is at the angle vertex. If is in the interior of, and, then. bisects, and is called the bisector of Postulate 4.2 S ongruence Postulate If two angles and the included side of one triangle are congruent to corresponding parts of another triangle, then the two triangles are congruent. O T Postulate 4.3 SSS ongruence Postulate If three sides of one triangle are congruent respectively to three sides of another triangle, then the two triangles are congruent. I V G J S

5 Theorem 4.4 onverse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. b c 2 2 If a + b = c 2, then is a right angle and is a right triangle. a xample: Given the diagram with congruent segments marked, prove.

There are three ways to classify triangles based on sides

There are three ways to classify triangles based on sides Unit 4 Notes: Triangles 4-1 Triangle ngle-sum Theorem ngle review, label each angle with the correct classification: Triangle a polygon with three sides. There are two ways to classify triangles: by angles