# ACT Triangles

```SAT / ACT GEOMETRY
TRIANGLES
The base angles of an isosceles triangle are equal
Example 1 In a triangle RST, if RS = ST and m (∠T) = 70&deg;, what is the
value, in degrees, of angle S?
Solution
RS = ST ⇒ m (∠R) = m (∠T) = 70&deg;;
m (∠S) = 180&deg; − (70&deg; + 70&deg;) = 40&deg;.
Example 2 In the figure, PQRS is a square and RST is an equilateral
triangle. Find the value of x.
Solution:
m (∠TSQ) = 90&deg; – m (∠RST) = 90&deg; – 60&deg; = 30&deg;
180&deg; − 30&deg;
ππ = ππ ⇒ π₯ = m (∠STQ) = m (∠SQT) =
= 75&deg;
2
ΜΜΜΜ and ππ
ΜΜΜΜ are angle bisectors. If m (∠P) =
Example 3 In the figure, ππ
80&deg;, how many degrees in ∠QSR?
Solution
m (∠QSR) = 180&deg; − π(∠πππ ) − π(∠πππ)
1
1
= 180&deg; − π(∠π ) − π(∠π)
= 180&deg; −
= 180&deg; −
2
1
2
1
2
1
2
[π(∠π) + π(∠π )]
[180&deg; − π(∠π)]
= 180&deg; − (180&deg; − 80&deg;)
2
= 180&deg; − 50&deg;
= 130&deg;
Example 4
Solution
Example 5 If the angles of a triangle are in the ratio of 2:3:5, what is
the measure of the smallest angle?
Solution
Let 2x&deg; be the smallest angle, then the other two angles would be 3x&deg;
and 5x&deg; other two angles in degrees.
2x + 3x + 5x = 180&deg;
10x = 180&deg;
x = 18&deg;
2x = 36&deg;
Example 6
Solution
Example 7 In a right triangle PQR, RH and RM are the altitude and the
median to the hypotenuse. If m (∠Q) = 32&deg;, find m (∠ HRM).
Solution In a right triangle, the median to the hypotenuse equals halve
the hypotenuse. Therefore, RM = PM = QM, and βRMQ is an isosceles
triangle. Hence, m (∠πππ) = 32&deg;. In triangle RMQ, ∠RMH is an
exterior angle. Thus,
m (∠RMH) = m (∠πππ) + π (∠πππ ) =32&deg; + 32&deg; = 64&deg;
m (∠ HRM) = 90&deg; – m (∠RMH) = 90&deg; – 64&deg; = 26&deg;
```