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

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SAT Math › How to find an angle in an acute / obtuse isosceles triangle

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The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?

CORRECT

Explanation

Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

The base angle of an isosceles triangle is five more than twice the vertex angle. What is the base angle?

$73$

CORRECT

$34$

$47$

$62$

$55$

Explanation

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let $x$ = the vertex angle and $2x+5$ = the base angle

So the equation to solve becomes $x+(2x+5)+(2x+5)=180$

Thus the vertex angle is 34 and the base angles are 73.

Triangle ABC has angle measures as follows:

$\dpi{100} \small m\angle ABC=4x+3$

$\dpi{100} \small m\angle ACB=2x+6$

$\dpi{100} \small m\angle BAC=3x$

What is $\dpi{100} \small m\angle BAC$ ?

CORRECT

Explanation

The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation $\dpi{100} \small 4x+3+2x+6+3x=180$

After combining like terms and cancelling, we have $\dpi{100} \small 9x=171\rightarrow x=19$

Thus $\dpi{100} \small m\angle BAC=3x=57$

In an isosceles triangle, the vertex angle is 15 less than the base angle. What is the base angle?

CORRECT

Explanation

Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = base angle and = vertex angle

So the equation to solve becomes

Thus, 65 is the base angle and 50 is the vertex angle.

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?

CORRECT

Explanation

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:

$\frac{x+y}{2}=55^o$