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Chapter 4 Review Booklet By Tim White
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Table of Contents: I.D. and classifying triangles by angles and sides Apply angle sum and exterior angle sum theorems Solving problems involving triangles Prove triangles are congruent by S.S.S and S.A.S. Prove triangles are congruent by A.S.A and A.A.S. Prove right triangles are congruent Solving problems with Isosceles triangles Coordinate proofs
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Classifying triangles by angles and sides Angles: There are three ways to classify a triangle by the measure of its angles. Acute triangle: All angles have to be acute (less than 90 degrees)
Obtuse Triangle: Has 1 obtuse angle
Right Triangle: Triangle that has one angle equal to 90 degrees
Sides: Scalene: No sides are congruent
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Isosceles Triangle: Triangle with 2 sides equal to each other
Equiangular/Equilateral triangles: Have both congruent sides and angles Examples: 1. What kind of triangle is this
Right because it has 1 angle equal to 90 degrees 2. What kind of triangle is this?
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Isosceles because both of the legs of the triangle are congruent
Applying Angle sum theorem and Exterior angle sum theorem Angle sum Theorem: Sum of the measures of a triangle is equal to 180 degrees Third angle theorem: If 2 angles of a triangle are congruent to 2 corresponding angles on another triangle then the third angles are congruent
Exterior angles:
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Every vertex has two exterior angles The sum of angle 1 and angle 2 would equal 180 degrees because they are a linear pair Remote interior angles:
The sum of the two remote interior angles equals the measure of the exterior angle
Example:
Corollary: Acute angles of a right triangle are complementary
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The measure of all 3 angles would equal 180 Simply proven by subtracting 90 from 180 and what you have left is what the other 2 angles have to equal. Also in any triangle there can’t be more than one right angle or one obtuse angle. Otherwise it goes over 180 degrees
Solving Problems Involving triangles
If 2 triangles corresponding parts are congruent then the triangles are congruent. Examples: 1. Triangle GTX is congruent to triangle ABC GT= 6x+4 AB= 5x +7 Find x 6x+4=5x+7 X=3
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2. Triangle Cat is congruent to triangle Dog Measure of Angle CAT= 55 Measure of Angle Dog= 6x – 5 Find measure of angle Dog 6x-5=55 6x=60 X=10
Side-Side-Side and Side-Angle-Side SSS or side-side-side is a postulate stating if a triangle’s sides are congruent to corresponding sides of another triangle then the triangles are congruent. Example: Given: Segment AC and AB are congruent and D is the midpoint of segment BC
Prove triangle ACD is congruent to Triangle ABD
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SAS or Side-Angle-Side is another postulate stating that if two triangles have two corresponding congruent sides and the included angles are congruent then the triangles are congruent Example
Given: C is the midpoint of segment BD Segment AC and EC are congruent
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Prove: Triangle ABC is congruent to triangle DEC
Proving triangles congruent by ASA and AAS ASA or Angle-Side-Angle is a postulate stating that if a triangle has 2 angles congruent to 2 angles of another triangle and the included sides are congruent then the triangles are congruent. Example: Given: Angle 4 and angle 6 are congruent B is the midpoint of segment AD Prove: triangle ACB is congruent to triangle DEB
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Angle-Angle-Side or AAS: IF two angles and a non included side of a triangle are congruent to another triangles coresponding parts then the triangles are congruent. Example:
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Given: Angle A is congruent to Angle E and angle D is congruent to angle C B is the midpoint of CD Prove: Triangle ACB is congruent to triangle EDB
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Prove Right Triangles are Congruent Parts of a right triangle:
Hypotoneuse leg theorem:
Leg Angle Theorem:
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Leg-Leg Theorem:
Hypotenuse-angle theorem:
Example: Given: Segment AC is congruent to Segment CB. Both Tiangles ACD and BCD are right triangles. Prove: Triangle ACD is congruent to Triangle BCD
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Solve problems with Isosceles triangle Isosceles has two congruent sides Parts of the isosceles triangle:
Isosceles Triangle theorem: If the opposite the legs of an Isosceles triangle are congruent then the base angles are congruent. Example:
2x+y=70 -2x+y=40 2y=110 Y=55
-2(55) + y = 40 -110 + y = 40 y=150
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Coordinate proofs Steps: 1. 2. 3. 4.
use origin as a vertex of the triangle place at least 1 side of triangle of an axis jeeo triangle in first quadrant if possible use coordinates that make computations simple
Example: Make a equlateral triangle that x is b distance from origin Label cordinates
F(0,0) E(B,0) D(B,B)