Lesson 4.3 and 4.4 Proving Triangles are Congruent

Lesson 4.3 and 4.4 Proving Triangles are Congruent p. 212

Learning Target I can list the conditions (SAS, SSS) to prove triangles are congruent. I can identify and use reflexive, symmetric and transitive property in my proof.

How To Find if Triangles are Congruent • Two triangles are congruent if they have: • exactly the same three sides and • exactly the same three angles. • But we don't have to know all three sides and all three angles ...usually three out of the sixis enough. • There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS (side, side, side) If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. SSS stands for "side, side, side“ and means that we have two triangles with all three sides equal. For example: is congruent to:

2. SAS (side, angle, side) If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal. For example: is congruent to:

3. ASA (angle, side, angle) If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. ASA stands for "angle, side, angle“ and means that we have two triangles where we know two angles and the included side are equal. For example: is congruent to:

4. AAS (angle, angle, side) If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. AAS stands for "angle, angle, side“ and means that we have two triangles where we know two angles and the non-included side are equal. For example: is congruent to:

5. HL (hypotenuse, leg) HL applies only to right angled-triangles! HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs") and

5. HL (hypotenuse, leg) If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent. • It means we have two right-angled triangles with • the same length of hypotenuseand • thesame length for one of the other two legs. • It doesn't matter which leg since the triangles could be rotated. • For example: is congruent to

Caution ! Don't Use "AAA" ! Without knowing at least one side, we can't be sure if two triangles are congruent.. AAA means we are given all three angles of a triangle, but no sides. This is not enough information to decide if two triangles are congruent! Because the triangles can have the same angles but be different sizes: For example: is congruent to

Goal 2 Proving Triangles are Congruent E B D A F C DEF DEF ABC DEF ABC ABC DEF JKL ABC JKL If  , then  . L If  and  , then . J K You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. THEOREM Theorem 4.4Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles

Using the SAS Congruence Postulate Prove that AEBDEC. 1 2 1 2 Statements Reasons AE  DE, BE  CE Given 1  2Vertical Angles Theorem 3 AEBDEC SAS Congruence Postulate

Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make DRAcongruent to DRG. You design the window so that DRAG and RARG. D A G R GIVEN DRAG RARG DRADRG PROVE MODELING A REAL-LIFE SITUATION Can you conclude that DRADRG? SOLUTION

Proving Triangles Congruent GIVEN RARG DRADRG PROVE 1 2 6 3 4 5 Statements Reasons Given DRAG If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. Right Angle Congruence Theorem DRADRG DRAG Given RARG DRDR Reflexive Property of Congruence SAS Congruence Postulate DRADRG D A R G

Given: SP  QR; QP  PRProve  SPQ  SPR S Statements Reasons 1. Given 1. SP  QR; QP  PR 2. QPS and RPS are right ’s. 2. Def. of  3. QPS  PRS 3. Rt.   Thm. 4. SP  SP 4. Reflexive POC 5.  SPQ  SPR 5. SAS  Post. Q R P

Pair-share Work on classwork on “Congruence Triangle” Sage and Scribe on #21 to #24

Lesson 4.3 and 4.4 Proving Triangles are Congruent