Chapter 3.7 Angle-Side Theorems. Erin Sanderson Mod 9.

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Presentation transcript:

Chapter 3.7 Angle-Side Theorems. Erin Sanderson Mod 9.

Objective. This section will teach you how to apply theorems relating to the angle measure and side lengths of triangles. Triangle =D

Theorem 20. If two sides of a triangle are congruent, the angles opposite the sides are congruent. (If, then.)

But Why? Statement.Reason A A Given 2.Reflexive Property 3.SAS (1,2,1) 4.CPCTC Given: Prove: A B C

Theorem 21; the Reverse. If two angles of a triangle are congruent, the sides opposite the angles are congruent. (if, then.)

How Come? Statement.Reason Given 2.Reflexive Property 3.ASA (1,2,1) 4.CPCTC Given: Prove: M E G

How Do I know if a is Isosceles? 1.If at least two sides of a triangle are congruent, the triangle is isosceles. 2.If at least two angles of a triangle are congruent, the triangle is isosceles.

The Inverses Also Work... If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Basically; This means that the longest side is across from the largest angle and the shortest side is across from the smallest angle.

It Would Kind of Look Like... LONGER LARGER SHORTERSHORTER SMALLERSMALLER That.

This means... Equilateral triangles are also equiangular because all of the sides are congruent, thus all of the angles are congruent.

Sample Problems. Statement.Reason. 1.ACDE is a square. 2.B bisects Given. 2.Given 3.All sides of a square are cong. 4.If a line is bisected, it is divided into 2 cong. lines 5.All angles of a square are cong. 6.SAS (3,4,5) 7.CPCTC 8.If sides, then angles AB C DE Prove: Given: ACDE is a square. B bisects.

#2 9x-72x+40 6x-12 A B C Given: Angle measures as shown; ABC is isosceles. Find: The measure of angle A. Since you know that B C, you can say that x+40=9x-72 8x=112 x=14 Then, you can substitute 14 in for the x in A. 6(14)-12 The answer is 72.

Now, do some on your own U Q R S T Given: QR ST; UR US Prove: QUS TUR

F GD E Given: F; GE ED Prove: EF bisects GFD

Answers. Statement.Reason. 1.QR ST; UR US 2.QS RT QUS TUR 1.Given 2.Addition 3.If sides, then angles 4.SAS (1,2,3)

And another… Statement.Reason 1.F; GE ED 2.GF FD 3. EGF EDF 4. EGF EDF 5.GFE DFE 6.EF bisects GFD 1.Given 2.Radii of a circle are congruent. 3.If sides, then angles. 4.SAS (1,2,3) 5.CPCTC 6.If a ray divides an angle into 2 congruent angles, the ray bisects the angle

Works Cited Geometry for Enjoyment and Challenge. New Edition. Evanston, Illinois: McDougal Littell, “Isosceles Triangle Proofs.” Math Warehouse. 29 May