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## Example Questions

### Example Question #1 : Calculating Whether Acute / Obtuse Triangles Are Congruent

Is it true that ?

Suppose you want to answer this question, and you know that and . Which of the following additional facts would help you to answer this question one way or the other?

**Possible Answers:**

None of these statements would be sufficient to answer the question.

**Correct answer:**

If you know either that or , you have three congruencies - two sides and a nonincluded angle. This is not enough to establish triangle congruence,

If you know that , this, along with the other two statements, establishes that ; all this proves is that both triangles are isosceles.

If you also know that , however, the three statements together make three side congruencies, setting up the Side-Side-Side criterion for triangle congruence.

### Example Question #1 : Calculating Whether Acute / Obtuse Triangles Are Congruent

Note: Figure NOT drawn to scale

Refer to the above diagram. Which of the following statements is NOT a consequence of the fact that ?

**Possible Answers:**

is the midpoint of

bisects

is an altitude of

is an equilateral triangle

is a right angle

**Correct answer:**

is an equilateral triangle

We use the congruence of corresponding sides and angles of congruent triangles to prove four of these statements:

, so bisects .

, so is the midpoint of .

, and, since they form a linear pair, they are supplementary - therefore, they are right angles. Also, by definition, this makes an altitude.

However, nothing in this congruence proves that is congruent to the other two sides of (which are congruent). The correct statement to exclude is that is equilateral.

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