Geometry - Prove theorems about triangles.

Which of the following theorems is used only to prove the congruence of two right triangles?

SSS Theorem

SAS Theorem

ASA Theorem

HL Theorem

Explanation

While all of these theorems can prove two triangles to be congruent the Hypotenuse-Leg Theorem (HL) is the only theorem out of these that can only prove two right triangles to be congruent. This theorem states that if two right triangles have one congruent leg and a congruent hypotenuse then they are congruent.

True or False: The SAS Theorem, ASA Theorem, SSS Theorem, and AA Theorem are all theorems that prove triangles to be congruent.

True

False

Explanation

Three of these theorems do prove triangles to be congruent; SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and SSS (Side-Side-Side). The AA (Angle-Angle) Theorem states that two triangles with two congruent, corresponding angles are similar NOT necessarily congruent.

Fill in the blanks to the proof below. We are proving that the base angles of isosceles triangles, , are congruent. This is a proof of the isosceles triangle theorem.

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$\overline{AB} \cong \overline{BA}$ , similar, angle bisector, SAS

$\overline{AC}\cong \overline{BC}$ , congruent, angle bisector, SAS

$\overline{AC}\cong \overline{BC}$ , congruent, midpoint, ASA

Explanation

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Consider the following theorem: If two triangles have two pairs of congruent corresponding sides and their included angle is congruent, then these two triangles are congruent. When proving this theorem, what information do we assume to begin the proof?

Assume that two triangles have a pair of corresponding congruent angles

Assume that two triangles have two pairs of corresponding congruent sides

Assume that two triangles have two pairs of corresponding congruent sides and their included angle is congruent

Assume that the two triangles are congruent

Explanation

When theorems are presented to you in an “if-then” format, you always assume the entire “if” part. This allows you to construct an argument that you will be able to prove to be true using the given information.

Which of the following is the altitude rule?

The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.

The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the leg.

The altitude to the leg of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.

The altitude to the leg of a right triangle is the mean proportional between the segments into which it divides the leg.

Explanation

This can be represented by the following equation

What is the first step to any proof?

State the given

Write your conclusion

There is no set first step

Draw a picture

Explanation

In order to prove anything, we must first state the given information. This will allow us to move forward with the proof and make different connections based on the given information.

Prove that if one side of a triangle is longer than another side of the same triangle, then the angle opposite the longer side will be greater than the angle opposite the shorter side. Use the information that $\overline{AB }< \overline{AC}$ for triangle .

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Proof:

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Proof:

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Proof:

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Explanation

Follow the detailed proof below.

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True or False: The Base Angles Converse states that if two base angles are congruent, then their opposite sides are congruent.

True

False

Explanation

Since the Base Angles Theorem states that if two adjacent sides are congruent then their opposite angles are congruent, then its converse assumes that the two base angles are congruent and then their opposite sides are congruent.

The triangle inequality theorem states that if you have a triangle then

Fill in the blanks for the proof of the theorem below using triangle

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To begin this proof we first must let there exist the triangle , where the length of $\overline{DC}\cong \overline{AC}$ and $\overline{AC}$ is a shared side between the two triangles.

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isosceles triangle, $\angle DAC< \angle DAB$

equilateral triangle, $\overline{AC}< \overline{CB}$

obtuse triangle, $\angle DAC< \angle DAB$

Explanation

Follow the detailed proof below for an explanation.

To begin this proof we first must let there exist the triangle , where the length of $\overline{DC}\cong \overline{AC}$ and $\overline{AC}$ is a shared side between the two triangles.