Using triangle congruence.

USING TRIANGLE CONGRUENCE

a very common way of showing that two segments are congruent is by looking

them as corresponding angles of congruent

triangles

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Side-Angle Relations in a Triangle• consider an isosceles triangle POM

that OP OM O

P N M

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Side-Angle Relations in a Triangle

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TheoremIsosceles Triangle

Theorem if two sides of a triangle are congruent, then the angles opposite those side are congruent

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Side-Angle Relations in a Triangle• Prove that an equilateral triangle

ABC is also equiangular Statement Reason AB BC definition of an equilateral

triangle

A C Isosceles triangle theorem

AB AC Definition of an equilateral triangle

B C isosceles triangle theorem

A B C Transitive property

∆ABC is equiangular. Def. of an equiangular angle



TheoremConverse of Isosceles

Triangle Theorem if two angles of a triangle are congruent, then the sides opposite those angles are congruent


Side-Angle Relations in a TriangleStatement Reason A C definition of an

A Bequiangular angle

AC BC Converse ofAB BC Isosceles Triangle

Theorem



Statement ReasonAB BC AC Transitive

property

∆ABC is equilateral Definition of an equilateral

triangle


Inequalities in a Triangle

• Is m C > m B ?Actual measurements shows that the

statement is true, but there is a need to reason out why this is so

A

B C



Extend AC to a point D such that AB AD

We now have an isosceles triangle.



Theorem If two sides of a triangle are not congruent, then the angles opposite these two sides are not congruent, and the larger angle is opposite the longer side



In ∆RSP RS = 35 RP = 31 PS = 521. Which is the largest scale?

2. Which is the smallest angle?Solution:

3. The largest angle is R 4. The largest angle is S



Theorem if two angles of a triangle are not congruent, then the sides opposite these two angles are not congruent, and the longer side is opposite the larger angle

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In the accompanying figure, OP = OM, m OPQ =145, and m POM =110

What is the longest side of ∆OPM ?Solution:Since the POMIs the largest angleSo PM is the longest side of ∆OPM Free powerpoint template: www.brainybetty.com


Congruence of right triangles



In any right triangle,The side opposite

of the right triangle is called the hypotenuseThe two others are Legs



TheoremLL congruence theorem

if the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent



Since all right triangles are congruent then

R M. thus, by ASA Congruence Postulate,

We have ∆ORS ∆LMN

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TheoremLA congruence

Theoremif a leg and an acute

angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.

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Since all right angles are congruent, then T Y. Thus, by SAA congruence postulate, we have

∆STU ∆XYZ



TheoremHyA Congruence Theorem

If an acute angle and the hypotenuse of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.


HyA Congruence Theorem



TheoremHyL Congruence

theorem if a leg and the

hypotenuse of one right triangle are congruent to a corresponding leg and the hypotenuse of another right triangle, then the triangles are congruent.



Extend ray DE to a point G such that GE AB. By SAS congruence Postulate, we have ∆ ABC ∆GEF.

we get AC GF


Key Expressions

HyA Congruence theoremHyL Congruence theoremLA Congruence theoremLL Congruence theoremCongruent trianglesEquilateral triangle

HypotenuseLegscongruence

Using triangle congruence.

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