Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

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

Parallel 2

Refer to the above diagram. \(\displaystyle \angle ABV \cong \angle DCV\).

True or false: From the information given, it follows that \(\displaystyle \bigtriangleup AVB \sim \bigtriangleup CVD\).

Possible Answers:

False

True

Correct answer:

False

Explanation:

The given information is actually inconclusive.

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar. Therefore, we seek to prove two of the following three angle congruence statements:

\(\displaystyle \angle AVB \cong \angle CVD\)

\(\displaystyle \angle ABV \cong \angle CDV\)

\(\displaystyle \angle BAV \cong \angle DCV\)

\(\displaystyle \angle AVB\) and \(\displaystyle \angle CVD\) are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, \(\displaystyle \angle AVB \cong \angle CVD\)

\(\displaystyle \angle ABV \cong \angle DCV\), but this is not one of the statements we need to prove. Also, without further information - for example, whether \(\displaystyle m\) and \(\displaystyle n\) are parallel, which is not given to us - we have no way to prove either of the other two necessary statements. 

The correct response is "false".

Example Question #27 : Triangles

\(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\) are both isosceles triangles;

\(\displaystyle m \angle B = 120 ^{\circ }\)

\(\displaystyle m \angle F = 30 ^{\circ }\)

True or false: from the given information, it follows that \(\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF\).

Possible Answers:

True

False

Correct answer:

False

Explanation:

As we are establishing whether or not \(\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF\), then \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) correspond respectively to \(\displaystyle D\)\(\displaystyle E\), and \(\displaystyle F\).

\(\displaystyle \bigtriangleup DEF\) is an isosceles triangle, so it must have two congruent angles. \(\displaystyle \angle F\) has measure \(\displaystyle 30^{\circ }\), so either \(\displaystyle \angle D\) has this measure, \(\displaystyle \angle E\) has this measure, or \(\displaystyle \angle D \cong \angle E\). If we examine the second case, it immediately follows that \(\displaystyle \angle B \ncong \angle E\). One condition of the similarity of triangles is that all pairs of corresponding angles be congruent; since there is at least one case that violates this condition, it does not necessarily follow that  \(\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF\). This makes the correct response "false".

Example Question #471 : Intermediate Geometry

\(\displaystyle \bigtriangleup ABC\) is an equilateral triangle; \(\displaystyle \bigtriangleup DEF\) is an equiangular triangle.

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

Possible Answers:

False

True

Correct answer:

True

Explanation:

As we are establishing whether or not \(\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF\), then \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) correspond respectively to \(\displaystyle D\)\(\displaystyle E\), and \(\displaystyle F\).

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 \(\displaystyle 60^{\circ }\) ). It follows that all angles of both triangles measure \(\displaystyle 60^{\circ }\).

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

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

Given: \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\) such that 

\(\displaystyle \angle A \cong \angle B\)

\(\displaystyle \angle C \cong \angle F\)

\(\displaystyle \angle D \cong \angle E\)

Which statement(s) must be true?

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

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

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 \(\displaystyle 180^{\circ }\), so

\(\displaystyle m \angle A + m \angle B + m \angle C = 180^{\circ }\)

\(\displaystyle m \angle A + m \angle B + m \angle C - m \angle C = 180^{\circ } - m \angle C\)

\(\displaystyle m \angle A + m \angle B = 180^{\circ } - m \angle C\)

Also,

\(\displaystyle m \angle A = m \angle B\),

so

\(\displaystyle m \angle A + m \angle A = 180^{\circ } - m \angle C\)

\(\displaystyle 2 \cdot m \angle A = 180^{\circ } - m \angle C\)

\(\displaystyle \frac{2 \cdot m \angle A }{2}= \frac{180^{\circ } - m \angle C}{2}\)

\(\displaystyle m \angle A = \frac{180^{\circ } - m \angle C}{2}\)

By similar reasoning, it holds that

\(\displaystyle m \angle D = \frac{180^{\circ } - m \angle F}{2}\)

Since \(\displaystyle m \angle F = m \angle C\), by substitution,

\(\displaystyle m \angle D = \frac{180^{\circ } - m \angle C}{2}\)

Therefore, 

\(\displaystyle m \angle A = m \angle D\),

or \(\displaystyle \angle A \cong \angle D\)

This, along with the statement that \(\displaystyle \angle C \cong \angle F\), 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 

\(\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF\).

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).

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:

\(\displaystyle 100\)

\(\displaystyle 40\)

\(\displaystyle 20\)

\(\displaystyle 60\)

\(\displaystyle 150\)

Correct answer:

\(\displaystyle 60\)

Explanation:

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

Let \(\displaystyle x\) = the base angle and \(\displaystyle 2x\ +\ 20\) = vertex angle

So the equation to solve becomes \(\displaystyle x\ +\ x\ +\ 2x\ +\ 20 = 180\)

or

 \(\displaystyle 4x\ +\ 20=180\)

so the base angle is \(\displaystyle 40\) and the vertex angle is \(\displaystyle 100\) and the difference is \(\displaystyle 60\).

Example Question #2 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An ssosceles triangle has interior angles of \(\displaystyle 32\) degrees and \(\displaystyle 74\) degrees. Find the missing angle. 

Possible Answers:

\(\displaystyle 32^\circ\) 

\(\displaystyle 15^\circ\)

\(\displaystyle 74^\circ\) 

\(\displaystyle 90^\circ\) 

\(\displaystyle 37^\circ\) 

Correct answer:

\(\displaystyle 74^\circ\) 

Explanation:

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

Thus, the solution is:

\(\displaystyle 32+74=106\)

\(\displaystyle 180-106=74\)

\(\displaystyle Check: 74+74+32=180\)

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

The largest angle in an obtuse isosceles triangle is \(\displaystyle 98\) degrees. Find the measurement of one of the two equivalent interior angles. 

Possible Answers:

\(\displaystyle 90^\circ\) 

\(\displaystyle 41^\circ\) 

\(\displaystyle 2^\circ\)

\(\displaystyle 82^\circ\) 

\(\displaystyle 42^\circ\) 

Correct answer:

\(\displaystyle 41^\circ\) 

Explanation:

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

Thus, the solution is:

\(\displaystyle 180-98=82\)

\(\displaystyle 82\div2=41\)

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 \(\displaystyle 28\) degrees. Find the measurement of the obtuse angle. 

Possible Answers:

\(\displaystyle 56^\circ\) 

\(\displaystyle 124^\circ\) 

\(\displaystyle 99^\circ\)

\(\displaystyle 134^\circ\) 

\(\displaystyle 112^\circ\) 

Correct answer:

\(\displaystyle 124^\circ\) 

Explanation:

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

Thus, the solution is:

\(\displaystyle 28+28=56\)

\(\displaystyle 180-56=124\)

Example Question #2 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

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

Possible Answers:

\(\displaystyle 32^\circ\) 

\(\displaystyle 4^\circ\) 

\(\displaystyle 26^\circ\) 

\(\displaystyle 13^\circ\) 

\(\displaystyle 10^\circ\) 

Correct answer:

\(\displaystyle 13^\circ\) 

Explanation:

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

The solution is:

\(\displaystyle 180-154=26\)

However, \(\displaystyle 26\) degrees is the measurement of both of the acute angles combined.

Each individual angle is \(\displaystyle 26\div2=13\).

Example Question #1 : 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 \(\displaystyle 53\) degrees. Find the missing angle. 

Possible Answers:

\(\displaystyle 74^\circ\) 

\(\displaystyle 62^\circ\) 

\(\displaystyle 75^\circ\) 

\(\displaystyle 42^\circ\)

\(\displaystyle 53^\circ\) 

Correct answer:

\(\displaystyle 74^\circ\) 

Explanation:

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

The solution is:

\(\displaystyle 53+53=106\)

\(\displaystyle 180-106=74\) 

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