KA-0152Angles on lines, in triangles, in polygons
3 points, difficulty 1 of 3Level 7–8about 75sIn a triangle, one angle measures 30 degrees. Of the other two angles, one is exactly twice the other. What is the largest angle in the triangle?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
The three angles add to 180 degrees, so the two unknown ones add to 150.
2The strategy
Call the smaller unknown angle x. Then the other is 2x, and x + 2x = 150.
3The full solution
3x = 150, so x = 50 and 2x = 100. The angles are 30, 50 and 100, which sum to 180.
Solution
The reliable way
The angles of a triangle sum to 180 degrees, so the two unknown angles together make 180 - 30 = 150. Writing the smaller as x makes the larger 2x, so x + 2x = 3x = 150 and x = 50. The two unknown angles are therefore 50 and 100, and the largest angle in the triangle is 100 degrees. Check: 30 + 50 + 100 = 180. The transferable idea: naming the smaller share x keeps an uneven split from collapsing into an even one.
The elegant way
The remainder of 150 splits into three equal shares of 50; the larger angle takes two of them, giving 100.
Why this is on the test: Naming the smaller quantity x is what keeps a one-to-two split from collapsing into an even one.