Geometry Problems: Angles, Triangles, Perimeter
Basic facts you need to know
Complementary
angles add to 90°,
same as a right
angle.
Supplementary
angles add to 180°,
same as a straight
line angle.
Page 1 of 2
Complementary Angles add to 90°
The problem. Find two complementary
angles such that the measure of the first
angle is 15° less than twice the measure of
the second angle.
Supplementary Angles add to 180°
The problem. Find two supplementary
angles such that the measure of the larger
angle is 110° more than triple the measure of
the smaller angle.
1. Thinking about it. We have a clue about
the first angle but no clue about the second
angle.
1. Thinking about it. We have a clue about
the larger angle but no clue about the smaller
angle.
2. The dictionary (write it out!).
Let 𝑥 = the second angle’s measure.
Then 2𝑥 − 15 = the first angle’s measure.
2. The dictionary (write it out!).
Let 𝑥 = the smaller angle’s measure.
Then 3𝑥 + 110 = the larger angle’s measure.
3. The equation. 𝑥 + 2𝑥 − 15 = 90
The 90 is because of complementary angles.
3. The equation. 𝑥 + 3𝑥 + 110 = 180
The 180 is because of supplementary angles.
4. Solve the equation.
Combine like terms: 3𝑥 − 15 = 90.
+15 to each side: 3𝑥 = 105
Divide each side by 3: 𝑥 = 35.
4. Solve the equation.
Combine like terms: 4𝑥 + 110 = 180.
-110 to each side: 4𝑥 = 70
Divide each side by 4: 𝑥 = 17.5
(In this special case the decimal will be okay.)
5. Bring the solution back to the dictionary.
Let 𝑥 = the smaller angle’s measure = 17.5°.
Then 3𝑥 + 110 = the larger angle’s measure
= 3(17.5) + 110 → 162.5°
5. Bring the solution back to the dictionary.
Let 𝑥 = the second angle’s measure. = 35°
Then 2𝑥 − 15 = the first angle’s measure =
2(35) − 15 → 70 − 15 → 55°.
6. Answer the question they asked.
The first angle measures 55° and the second
angle measures 35°.
6. Answer the question they asked.
The angles measure 17.5° and 162.5°.
Quick check. Add: 17.5+162.5=180; that
Quick check. Add: 55+35=90; that agrees
agrees with the definition of supplementary
with the definition of complementary angles. angles.
Check your solution against the original problem: does it work? And always be sure to answer the question they asked!
D.R.S., 2/8/2016 1:12 AM
Geometry Problems: Angles, Triangles, Perimeter
Three angles of a triangle sum to 180°
The problem. The smallest angle of a triangle measures one-fifth as
much as the largest angle. The other angle measures ten degrees
more than one-half of the largest angle. Find the angles’
measurements.
1. Thinking about it. We have clues about the smallest angle and the
second angle. There aren’t any clues given about the largest angle
Page 2 of 2
Perimeter of a rectangle
The problem. A rectangular lot of land has a length that is 50 feet
shorter than three times its width. Its perimeter is 460 feet. What
are the dimensions of the lot?
1. Thinking about it. We have a clue about the length.
2. The dictionary (write it out!).
Let 𝑥 = the largest angle.
1
Then 𝑥 = the smallest angle.
2. The dictionary is a picture. Draw
it. Label the unknown width as x.
Label the length according to the clue
“50 feet shorter than the width.”
And
+ 10 = the second angle.
3. The equation. The key fact to know is that the three angles of a
1
1
triangle sum to 180° 𝑥 + 𝑥 + 𝑥 + 10 = 180
5
2
4. Solve the equation. Clear the fractions first. LCD of 5 and 2 is 10,
so multiply each term by 10: 10𝑥 + 2𝑥 + 5𝑥 + 100 = 1800
Combine like terms:
17𝑥 + 100 = 1800
Then -100 on each side:
17𝑥 = 1700
Div each side by 17:
𝑥 = 100
3. The equation. The key fact to
know is that perimeter is the distance
around the sides. Add up all the sides = the perimeter.
3𝑥 − 50 + 𝑥 + 3𝑥 − 50 + 𝑥 = 460
4. Solve the equation.
Combine like terms: 8𝑥 − 100 = 460
Then +100 on each side:
8𝑥 = 560
Div each side by 8:
𝑥 = 70
5
1
𝑥
2
5. Bring the solution back to your
diagram; plug in for x.
5. Bring the solution back to the dictionary.
Let 𝑥 = the largest angle = 100°
1
1
Then 5 𝑥 = the smallest angle = 5 (100) →20°
1
2
And 𝑥 + 10 = the second angle=
1
(100) +
2
10 →60°
Quick check. Yes, 160 + 70 +
160 + 70 = 460 feet.
Quick check. Yes, 100 + 20 + 60 = 180
6. Answer the question they asked. (Read carefully to put the
correct angle value in the correct box in the MyMathLab questions! )
6. Answer the question they asked. (Read carefully to put the correct
angle value in the correct box in the MyMathLab questions!)
D.R.S., 2/8/2016 1:12 AM