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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #18 : Acute / Obtuse Triangles

Triangle

Refer to the above diagram.

$45^{\circ } \leq m \angle 1 \leq 50^{\circ }$

$130^{\circ } \leq m \angle 3 \leq 140 ^{\circ }$

Which of the following could be a measure of $\angle 2$ ?

Possible Answers:

$85 ^{\circ }$

$80 ^{\circ }$

All of the other responses are correct.

$95 ^{\circ }$

$90 ^{\circ }$

Correct answer:

All of the other responses are correct.

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so

$\angle 3 = \angle 1 + \angle 2$

$\angle 2 = \angle 3 - \angle 1$

Therefore, the maximum value of $\angle 2$ is the least possible value of $\angle 1$ subtracted from the greatest possible value of $\angle 3$ :

$140^{\circ } - 45^{\circ } = 95 ^{\circ }$

The minimum value of $\angle 2$ is the greatest possible value of $\angle 1$ subtracted from the least possible value of $\angle 3$ :

$130^{\circ } - 50^{\circ } = 80 ^{\circ }$

Therefore,

$80^{\circ } \leq \angle 2 \leq 95^{\circ }$

Since all of the choices fall in this range, all are possible measures of $\angle 2$ .

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Example Question #19 : Acute / Obtuse Triangles

Find the angle measurement of .

Possible Answers:

102

112

Correct answer:

102

Explanation:

All the angles in a triangle must add up to 180 .

32+46+y=180

78+y=180

y=102

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Example Question #4 : How To Find An Angle In An Acute / Obtuse Triangle

Find the angle measurement of .

Possible Answers:

110

100

120

Correct answer:

110

Explanation:

All the angles in a triangle must add up to 180.

a+35+35=180

a+70=180

a=110

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

Find the angle measurement of .

Possible Answers:

115

105

Correct answer:

115

Explanation:

All the angles in a triangle must add up to 180 .

b+28+37=180

b+65=180

b=115

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Example Question #21 : Acute / Obtuse Triangles

An isosceles triangle has an angle whose measure is $70^{\circ }$ .

What could be the measures of one of its other angles?

(a) $40^{\circ }$

(b) $55^{\circ }$

Possible Answers:

(b) only

(a) or (c) only

(a) only

(a), (b), or (c)

Correct answer:

(a), (b), or (c)

Explanation:

By the Isosceles Triangle Theorem, an isosceles triangle has two congruent interior angles. There are two possible scenarios if one angle has measure $70^{\circ }$ :

Scenario 1: The other two angles are congruent to each other. The degree measures of the interior angles of a triangle total $180^{\circ }$ , so if we let be the common measure of those angles:

x + x + 70 = 180

2x+70 = 180

2x= 110

x = 55

This makes (b) a possible answer.

Scenario 2: One of the other angles measures $70^{\circ }$ also, making (c) a possible answer. The degree measure of the third angle is

180 - (70 + 70) = 180 - 140 = 40 ,

making (a) a possible answer. Therefore, the correct choice is (a), (b), or (c).

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Example Question #9 : How To Find An Angle In An Acute / Obtuse Triangle

One of the interior angles of a scalene triangle measures $54^{\circ }$ . Which of the following could be the measure of another of its interior angles?

Possible Answers:

$54^{\circ }$

$108^{\circ }$

$63^{\circ }$

$72^{\circ }$

$126^{\circ }$

Correct answer:

$108^{\circ }$

Explanation:

A scalene triangle has three sides of different measure, so, by way of the Converse of the Isosceles Triangle Theorem, each angle is of different measure as well. We can therefore eliminate $54^{\circ }$ immediately.

Also, if the triangle also has a $72^{\circ }$ angle, then, since the total of the degree measures of the angles is $180^{\circ }$ , it follows that the third angle has measure

$180 - (54+72) = 180 - 126 = 54 ^{\circ }$ .

Therefore, the triangle has two angles that measure the same, and $72^{\circ }$ can be eliminated.

Similarly, if the triangle also has a $63^{\circ }$ angle, then, since the total of the degree measures of the angles is $180^{\circ }$ , it follows that the third angle has measure

$180 - (54+63) = 180 - 117= 63 ^{\circ }$ .

The triangle has two angles that measure $63^{\circ }$ . This choice can be eliminated.

$126^{\circ }$ can be eliminated, since the third angle would have measure

$180 - (54+126) = 180 - 180= 0^{\circ }$ ,

an impossible situation since angle measures must be positive.

The remaining possibility is $108^{\circ }$ . This would mean that the third angle has measure

$180 - (54+108) = 180 - 162= 18^{\circ }$ .

The three angles have different measures, so the triangle is scalene. $108^{\circ }$ is the correct choice.

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Example Question #111 : Properties Of Triangles

Given: $\bigtriangleup ABC$ with $m \angle A = 20^{\circ }, m \angle B = 32^{\circ }$ . Locate on $\overline{AB}$ so that $\overrightarrow{CD}$ is the angle bisector of $\angle ACB$ . What is $m \angle CDB$ ?

Possible Answers:

$69^{\circ }$

$79^{\circ }$

$89^{\circ }$

$84^{\circ }$

$74^{\circ }$

Correct answer:

$84^{\circ }$

Explanation:

Angle bisector

Above is the figure described.

The measures of the interior angles of a triangle total $180^{\circ }$ , so the measure of $\angle ACB$ is

$m \angle ACB = 180^{\circ } -( m \angle A + m \angle B)$

$= 180^{\circ } -( 20^{\circ } + 32^{\circ })$

$= 180^{\circ } - 52^{\circ }$

$= 128^{\circ }$

Since $\overrightarrow{CD}$ bisects this angle,

$m \angle BCD= \frac{1}{2}m \angle ACB = \frac{1}{2} \cdot 128^{\circ } =64^{\circ }$

and

$m \angle CDB = 180^{\circ } -( m \angle BCD + m \angle B)$

$= 180^{\circ } -( 64 ^{\circ } + 32^{\circ } )$

$= 180^{\circ } -96^{\circ }$

$= 84^{\circ }$

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Example Question #21 : Acute / Obtuse Triangles

Given: $\bigtriangleup ABC$ with $m \angle B = 74^{\circ }$ . is located on $\overline{AB}$ so that $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ and forms isosceles triangle $\bigtriangleup BCD$ .

Give the measure of $\angle A$ .

Possible Answers:

$64^{\circ }$

Insufficient information is given to answer the question.

$42^{\circ }$

$74^{\circ }$

$53^{\circ }$

Correct answer:

$42^{\circ }$

Explanation:

If $\bigtriangleup BCD$ is isosceles, then by the Isosceles Triangle Theorem, two of its angles must be congruent.

Case 1: $m \angle B = m \angle BCD= 74^{\circ }$

Since $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ into two congruent angles, one of which must be $m \angle BCD$ ,

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 74^{\circ } = 148^{\circ }$

However, this is impossible, since $\angle ACB$ and $\angle B$ are two angles of the original triangle; their total measure is

$m \angle ACB+ m \angle B = 148^{\circ }+ 74 ^{\circ } = 222 ^{\circ } > 180 ^{\circ }$

Case 2: $m \angle B = m \angle BDC= 74^{\circ }$

Then, since the degree measures of the interior angles of a triangle total $180^{\circ }$ ,

$m \angle BCD= 180^{\circ } - ( \angle B + m \angle BDC )$

$= 180^{\circ } - (74^{\circ } + 74^{\circ } )$

$= 180 ^{\circ } - 148^{\circ }$

$= 32^{\circ }$

Since $\overrightarrow{CD}$ bisects $\angleACB$ $\angle ACB$ into two congruent angles, one of which must be $m \angle BCD$ ,

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 32^{\circ } = 64^{\circ }$

and

$m \angle A = 180^{\circ } - ( \angle B + m \angle ACB )$

$= 180^{\circ } - (74^{\circ } + 64^{\circ } )$

$= 180^{\circ } - 138^{\circ }$

$= 42^{\circ }$

Case 3: $m \angle BDC= m \angle BCD$

Then

$m \angle B = 180^{\circ } -(m \angle BDC+ m \angle BCD)$

$74= 180^{\circ } -(m \angle BCD+ m \angle BCD)$

$74= 180^{\circ } -2 \cdot m \angle BCD$

$2 \cdot m \angle BCD = 106^{\circ }$

$m \angle BCD = 53^{\circ }$

$m \angle ACB = 2 \cdot m \angle BCD = 2 \cdot 53^{\circ } = 106^{\circ }$

$m \angle A = 180^{\circ } - ( \angle B + m \angle ACB )$

$= 180^{\circ } - ( 74^{\circ }+ 106^{\circ } )$

$= 0^{\circ }$ , which is not possible.

Therefore, the only possible measure of $\angle A$ is $42^{\circ }$ .

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Example Question #111 : Properties Of Triangles

The interior angles of a triangle measure $x^{\circ } , \left ( x+ 26\right )^{\circ } , \left (2x+ 1\right )^{\circ }$ . Of these three degree measures, give the greatest.

Possible Answers:

$64\frac{1}{4} ^{\circ }$

This triangle cannot exist.

$77^{\circ }$

$77\frac{1}{2} ^{\circ }$

$103^{\circ }$

Correct answer:

$77\frac{1}{2} ^{\circ }$

Explanation:

The degree measures of the interior angles of a triangle total 180 degrees, so

x + (x+26) + (2x+ 1) = 180

x +x+ 2x + 26 + 1 = 180

4x+27 = 180

4x+27 - 27 = 180 - 27

4x = 153

$\frac{4x}{4} = \frac{153}{4}$

$x = 38 \frac{1}{4}$

One angle measures $38\frac{1}{4}^{\circ }$

The other two angles measure

$x+ 26 = 38\frac{1}{4} + 26 = 64\frac{1}{4} ^{\circ }$

and

$2x+ 1 = 2 \left$38\frac{1}{4}\right$+ 1 = 77 \frac{1}{2} ^{\circ }$ .

We want the greatest of the three, or $77 \frac{1}{2} ^{\circ }$ .

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Example Question #23 : Acute / Obtuse Triangles

$\bigtriangleup ABC$ is a right triangle with right angle $\angle A$ . is located on $\overline{AB}$ so that, when $\overline{CD}$ is constructed, isosceles triangles $\bigtriangleup ADC$ and $\bigtriangleup BDC$ are formed.

What is the measure of $\angle B$ ?

Possible Answers:

$67\frac{1}{2}^{\circ }$

$30^{\circ }$

$45^{\circ }$

$15^{\circ }$

$22\frac{1}{2}^{\circ }$

Correct answer:

$22\frac{1}{2}^{\circ }$

Explanation:

The figure referenced is below:

Right triangles

Since $\bigtriangleup ADC$ is an isosceles right triangle, its acute angles - in particular, $\angle ADC$ - measure $45^{\circ }$ each. Since this angle forms a linear pair with $\angle CDB$ :

$m \angle CDB = 180^{\circ } - m \angle ADC = 180^{\circ } - 45 ^{\circ } = 135^{\circ }$ .

$\bigtriangleup BDC$ is also isosceles, so, by the Isosceles Triangle Theorem, it has two congruent angles. Since $\angle CDB$ is obtuse, and no triangle has two obtuse angles:

$m \angle BCD = m \angle B$ .

Also, $\angle ADC$ is an exterior angle of $\bigtriangleup BDC$ , whose measure is equal to the sum of those of its two remote interior angles, which are the congruent angles $m \angle BCD = m \angle B$ . Therefore,

$m \angle ADC = m \angle BCD + m \angle B$

$45^{\circ }= m \angle B + m \angle B$

$45^{\circ }=2 m \angle B$

$m \angle B = 22\frac{1}{2}^{\circ }$

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SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #18 : Acute / Obtuse Triangles

Example Question #19 : Acute / Obtuse Triangles

Example Question #4 : How To Find An Angle In An Acute / Obtuse Triangle

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

Example Question #21 : Acute / Obtuse Triangles

Example Question #9 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #111 : Properties Of Triangles

Example Question #21 : Acute / Obtuse Triangles

Example Question #111 : Properties Of Triangles

Example Question #23 : Acute / Obtuse Triangles

All SSAT Upper Level Math Resources

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