using triangle congruence
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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
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Side-Angle Relations in a Triangle
TheoremConverse of Isosceles
Triangle Theorem if two angles of a triangle are congruent, then the sides opposite those angles are congruent
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Side-Angle Relations in a TriangleStatement Reason A C definition of an
A Bequiangular angle
AC BC Converse ofAB BC Isosceles Triangle
Theorem
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Side-Angle Relations in a Triangle
Statement ReasonAB BC AC Transitive
property
∆ABC is equilateral Definition of an equilateral
triangle
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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
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Inequalities in a Triangle
Extend AC to a point D such that AB AD
We now have an isosceles triangle.
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Inequalities in a 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
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Inequalities in a Triangle
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
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Inequalities in a Triangle
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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Inequalities in a Triangle
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
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Congruence of right triangles
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Congruence of right triangles
In any right triangle,The side opposite
of the right triangle is called the hypotenuseThe two others are Legs
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Congruence of right triangles
TheoremLL congruence theorem
if the legs of one right triangle are congruent to the legs of another right triangle, then the triangles are congruent
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Congruence of right triangles
Since all right triangles are congruent then
R M. thus, by ASA Congruence Postulate,
We have ∆ORS ∆LMN
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Congruence of right triangles
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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Congruence of right triangles
Since all right angles are congruent, then T Y. Thus, by SAA congruence postulate, we have
∆STU ∆XYZ
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Congruence of right triangles
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.
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HyA Congruence Theorem
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Congruence of right triangles
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.
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Congruence of right triangles
Extend ray DE to a point G such that GE AB. By SAS congruence Postulate, we have ∆ ABC ∆GEF.
we get AC GF
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Key Expressions
HyA Congruence theoremHyL Congruence theoremLA Congruence theoremLL Congruence theoremCongruent trianglesEquilateral triangle
HypotenuseLegscongruence
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