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SSAT Upper Level Math : How to find an angle in an acute / obtuse triangle

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Example Questions

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 : How to find an angle in an acute / obtuse triangle

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

All SSAT Upper Level Math Resources

Example Questions

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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