Lines, angles & triangles
These problems test a handful of angle relationships and triangle facts. The reliable move is to label every angle you can deduce from the figure, then write one equation from the relationship the question points to.
What College Board tests
Angles formed when a transversal crosses parallel lines, the triangle angle sum, the exterior-angle theorem, similar triangles (often created by a line parallel to one side), and the properties of isosceles and equilateral triangles.
The facts these rely on
Label first, then write one equation. Most of these questions collapse to "set two expressions equal" or "make the angles sum to ."
Five worked examples covering the core angle & triangle types. Read the approach, try it, then tap Show the full solution.
1 · Parallel lines and a transversal
In the figure, lines and are parallel, cut by a transversal. One angle measures and the corresponding angle measures . What is the value of ?
Approach Corresponding angles at parallel lines are equal. Set the two expressions equal and solve for .
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Answer: B
Why the other choices are wrong
A Solves with an arithmetic slip.
C Treats the angles as supplementary (summing to 180) by mistake.
D Uses the value of the angle (70) divided incorrectly.
2 · The triangle angle sum
The three angles of a triangle have measures , , and . What is the measure, in degrees, of the largest angle?
Approach All three angles add to . Set the sum equal to 180, solve for , then identify the largest (the term).
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Answer: C
Why the other choices are wrong
A Reports (the smallest angle).
B Reports the middle angle .
D Solves as if the largest were .
3 · Similar triangles
In triangle , segment is parallel to side , with on and on . If , , and , what is the length of ?
Approach Because , triangle is similar to triangle . Corresponding sides share the ratio , so is that fraction of .
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Answer: B
Why the other choices are wrong
A Uses the ratio instead of .
C Uses ratio (mistakes for half of ).
D Uses the ratio (the lower piece ).
4 · The exterior angle theorem
An exterior angle of a triangle measures . One of its two remote interior angles measures . What is the measure, in degrees, of the other remote interior angle?
Approach An exterior angle equals the sum of the two remote interior angles. So subtract the known interior angle from the exterior angle.
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Answer: C
Why the other choices are wrong
A Subtracts from 80 or mis-subtracts.
B Subtracts 45 from 100.
D Uses the supplement of 110 (which is the third interior angle, not asked).
5 · Isosceles triangle angles
In an isosceles triangle, the angle between the two equal sides (the vertex angle) measures . What is the measure, in degrees, of each base angle?
Approach The two base angles of an isosceles triangle are equal. Subtract the vertex angle from , then split the remainder evenly between the two equal base angles.
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Answer: C
Each base angle is . Check: . ✓
Why the other choices are wrong
A Assumes the base angles equal the vertex angle (only true if equilateral).
B Splits 100 instead of 140.
D Reports the combined without halving.