SHSAT: Angle relationships: write the total, then solve
Use straight-line, triangle, polygon and parallel-line angle facts to write and solve equations for unknown angles.
When angles are given as expressions with x, how do you find the angle the question wants?
Short answer
Identify the relationship, such as angles on a line adding to 180°, write it as an equation, solve for x, then substitute to find the angle asked for.
Know first: Solving linear equations, Basic angle vocabulary
Step 1 of 5
Every angle question hides a known total
Angle questions give you expressions such as 3x + 10 and expect you to know which angles add to what. Find the relationship, write it as an equation, and solve for x.
The SHSAT guide says formulas and definitions won't be supplied, so these totals have to be in your memory before test day.
| Situation | Relationship |
|---|---|
| Angles on a straight line | add to 180° |
| Angles around a point | add to 360° |
| Angles in a triangle | add to 180° |
| Interior angles of an n-sided polygon | add to 180°(n − 2) |
| Exterior angles of any polygon, one at each vertex | add to 360° |
| Exterior angle of a triangle | equals the sum of the two remote interior angles |
| Vertical angles | equal |
| Parallel lines cut by a transversal | corresponding and alternate angles are equal; same-side interior angles add to 180° |
An isosceles triangle has two equal base angles. A regular polygon has all angles equal.
Step 2 of 5
Write the total, solve, substitute
Worked example: Three angles of a triangle
The angles of a triangle measure (x + 15)°, 2x°, and (3x − 15)°. What is the largest angle?
- A30°
- B60°
- C75°
- D90°
- 1
Use the triangle total
(x + 15) + 2x + (3x − 15) = 180, so 6x = 180.
- 2
Solve
x = 30.
- 3
Substitute
The angles are 45°, 60° and 75°. They add to 180°, and the largest is 75°.
- 4
See why the other choices are there
30 is x, not an angle. 60 is the middle angle. 90 is 3x without the − 15.
Answer
75°
Find the wrong step
The interior angles of a convex polygon add to 1440°. How many sides does it have?
- 1
Use the rule: interior sum = 180°(n − 2).
- 2
Divide: 1440 ÷ 180 = 8.
- 3
So the polygon has 8 sides.
What went wrong
8 is n − 2, the number of triangles the polygon splits into, not the number of sides.
The fix
n − 2 = 8, so n = 10 sides. Check: 180 × 8 = 1440.
Step 3 of 5
Name the relationship before you calculate
Check yourself · Question 1
An isosceles triangle has a vertex angle of 48°. What is the measure of each base angle?
Answer: B
The two base angles are equal and share 180 − 48 = 132°. Each is 132 ÷ 2 = 66°.
Check yourself · Question 2
Lines p and q are parallel, and a transversal crosses both. Two same-side interior angles measure (3x + 10)° and (5x + 2)°. What is the measure of the angle (3x + 10)°?
Answer: C
Same-side interior angles add to 180°: 8x + 12 = 180, so x = 21. Then 3(21) + 10 = 73°. The other angle is 107°, and 73 + 107 = 180.
Trap answer: 22°
Using the wrong parallel-line rule
- Why it looks right
- Many parallel-line questions set two angles equal, so setting them equal feels automatic.
- Why it's wrong
- Same-side interior angles sit on the same side of the transversal between the lines. They are supplementary, not equal.
- The right answer
- 73°, from (3x + 10) + (5x + 2) = 180.
Check yourself · Question 3
Each interior angle of a regular polygon measures 160°. How many sides does it have?
Answer: C
Each exterior angle is 180 − 160 = 20°. The exterior angles add to 360°, so there are 360 ÷ 20 = 18 sides. Check: 180(18 − 2) = 2880, and 2880 ÷ 18 = 160.
Common mistake
I solve for x and pick the choice that matches it.
- Why it's tempting
- Solving the equation is the hardest part, so the moment you have x feels like the finish.
- Do this instead
- Reread the question after you find x. Substitute it into the expression for the angle asked, and check the angles fit the total.
Step 4 of 5
What to remember
Remember
- Know the totals: 180° on a line and in a triangle, 360° around a point and for exterior angles, 180°(n − 2) inside a polygon.
- With parallel lines, corresponding and alternate angles are equal, and same-side interior angles add to 180°.
- After solving for x, substitute and report the angle the question asks for.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which angle relationship gave you the equation, and which value did you substitute x into at the end?