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Published byClaribel Walton Modified over 9 years ago
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Lesson 4.3 and 4.4 Proving Triangles are Congruent
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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.
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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 six is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.
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1. SSS (side, side, side) SSS stands for "side, side, side“
and means that we have two triangles with all three sides equal. For example: is congruent to: If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
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2. SAS (side, angle, side) 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: 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.
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3. ASA (angle, side, angle) 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: 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.
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4. AAS (angle, angle, side) For example: is congruent to:
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: 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.
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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 HL applies only to right angled-triangles!
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5. HL (hypotenuse, leg) It means we have two right-angled triangles with the same length of hypotenuse and the same 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 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.
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Caution ! Don't Use "AAA" ! 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 Without knowing at least one side, we can't be sure if two triangles are congruent..
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Proving Triangles are Congruent
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 A B C Theorem 4.4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. D E F Symmetric Property of Congruent Triangles If , then ABC DEF J K L Transitive Property of Congruent Triangles If and , then JKL ABC DEF Goal 2
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Using the SAS Congruence Postulate
Prove that AEB DEC. 2 1 Statements Reasons AE DE, BE CE Given 1 1 2 Vertical Angles Theorem 2 AEB DEC SAS Congruence Postulate 3
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Proving Triangles Congruent
MODELING A REAL-LIFE SITUATION Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make DRA congruent to DRG. You design the window so that DR AG and RA RG. Can you conclude that DRA DRG ? D G A R SOLUTION GIVEN DR AG RA RG PROVE DRA DRG
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Proving Triangles Congruent
D GIVEN PROVE DRA DRG DR AG RA RG A R G Statements Reasons 1 Given DR AG If 2 lines are , then they form 4 right angles. DRA and DRG are right angles. 2 3 Right Angle Congruence Theorem DRA DRG 4 Given RA RG 5 Reflexive Property of Congruence DR DR 6 SAS Congruence Postulate DRA DRG
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Given: SP QR; QP PR Prove SPQ SPR
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.
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Pair-share Work on classwork on “Congruence Triangle”
Sage and Scribe on #21 to #24
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