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Intermediate Geometry : Triangles

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Example Question #1 : How To Find If Of Acute / Obtuse Isosceles Triangle Are Similar

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

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Example Question #2 : How To Find If Of Acute / Obtuse Isosceles Triangle Are Similar

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ such that

$\angle A \cong \angle B$

$\angle C \cong \angle F$

$\angle D \cong \angle E$

Which statement(s) must be true?

(a) $\bigtriangleup ABC \sim \bigtriangleup DEF$

(b) $\bigtriangleup ABC \cong \bigtriangleup DEF$

Possible Answers:

(a) but not (b)

(b) but not (a)

(a) and (b)

Neither (a) nor (b)

Correct answer:

(a) but not (b)

Explanation:

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ , so

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$m \angle A + m \angle B + m \angle C - m \angle C = 180^{\circ } - m \angle C$

$m \angle A + m \angle B = 180^{\circ } - m \angle C$

Also,

$m \angle A = m \angle B$ ,

$m \angle A + m \angle A = 180^{\circ } - m \angle C$

$2 \cdot m \angle A = 180^{\circ } - m \angle C$

$\frac{2 \cdot m \angle A }{2}= \frac{180^{\circ } - m \angle C}{2}$

$m \angle A = \frac{180^{\circ } - m \angle C}{2}$

By similar reasoning, it holds that

$m \angle D = \frac{180^{\circ } - m \angle F}{2}$

Since $m \angle F = m \angle C$ , by substitution,

$m \angle D = \frac{180^{\circ } - m \angle C}{2}$

Therefore,

$m \angle A = m \angle D$ ,

or $\angle A \cong \angle D$

This, along with the statement that $\angle C \cong \angle F$ , sets up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem.

Therefore, statement (a) must hold, but not necessarily statement (b).

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An isoceles triangle has a vertex angle that is twenty more than twice the base angle. What is the difference between the vertex and base angles?

Possible Answers:

150

100

Correct answer:

Explanation:

A triangle has 180 degrees. An isoceles triangle has one vertex angle and two congruent base angles.

Let = the base angle and $2x\ +\ 20$ = vertex angle

So the equation to solve becomes $x\ +\ x\ +\ 2x\ +\ 20 = 180$

$4x\ +\ 20=180$

so the base angle is and the vertex angle is 100 and the difference is .

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An ssosceles triangle has interior angles of degrees and degrees. Find the missing angle.

Possible Answers:

$32^\circ$

$15^\circ$

$74^\circ$

$37^\circ$

$90^\circ$

Correct answer:

$74^\circ$

Explanation:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of 180 degrees.

Thus, the solution is:

32+74=106

180-106=74

Check: 74+74+32=180

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Example Question #3 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

The largest angle in an obtuse isosceles triangle is degrees. Find the measurement of one of the two equivalent interior angles.

Possible Answers:

$42^\circ$

$2^\circ$

$90^\circ$

$82^\circ$

$41^\circ$

Correct answer:

$41^\circ$

Explanation:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of 180 degrees. Since this is an obtuse isosceles triangle, the two missing angles must be acute angles.

Thus, the solution is:

180-98=82

$82\div2=41$

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

The two equivalent interior angles of an obtuse isosceles triangle each have a measurement of degrees. Find the measurement of the obtuse angle.

Possible Answers:

$134^\circ$

$112^\circ$

$99^\circ$

$56^\circ$

$124^\circ$

Correct answer:

$124^\circ$

Explanation:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of 180 degrees.

Thus, the solution is:

28+28=56

180-56=124

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Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

In an obtuse isosceles triangle the angle measurements are, $x^\circ$ , $x^\circ$ , and $(10x-2)=128^\circ$ . Find the measurement of one of the acute angles.

Possible Answers:

$32^\circ$

$26^\circ$

$13^\circ$

$10^\circ$

$4^\circ$

Correct answer:

$13^\circ$

Explanation:

The solution is:

180-154=26

However, degrees is the measurement of both of the acute angles combined.

Each individual angle is $26\div2=13$ .

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Example Question #6 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

In an acute isosceles triangle the two equivalent interior angles each have a measurement of degrees. Find the missing angle.

Possible Answers:

$74^\circ$

$42^\circ$

$62^\circ$

$53^\circ$

$75^\circ$

Correct answer:

$74^\circ$

Explanation:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of 180 degrees. Since this is an acute isosceles triangle, all of the interior angles must be acute angles.

The solution is:

53+53=106

180-106=74

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Example Question #7 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

In an acute isosceles triangle the two equivalent interior angles are each degrees. Find the missing angle.

Possible Answers:

$70^\circ$

$62^\circ$

$140^\circ$

$13^\circ$

$78^\circ$

Correct answer:

$70^\circ$

Explanation:

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Example Question #8 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

The largest angle in an obtuse isosceles triangle is 138 degrees. Find the measurement of one of the equivalent interior angles.

Possible Answers:

$44^\circ$

$15^\circ$

$23^\circ$

$21^\circ$

$42^\circ$

Correct answer:

$21^\circ$

Explanation:

Isosceles triangles always have two equivalent interior angles, and all three interior angles of any triangle always have a sum of 180 degrees. Since this is an obtuse Isosceles triangle, the two missing angles must be acute angles.

Thus, the solution is:

180-138=42

$42\div2=21$

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Intermediate Geometry : Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find If Of Acute / Obtuse Isosceles Triangle Are Similar

Example Question #2 : How To Find If Of Acute / Obtuse Isosceles Triangle Are Similar

Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #3 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #6 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #7 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

Example Question #8 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

All Intermediate Geometry Resources

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