Level 3 · Challenge ← Back to lesson

Sides, Angles, and Shortest Distances: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

In triangle ABC, AC is longer than AB. To begin the cutoff proof, D is chosen on ray AC so AD equals AB. Why is D strictly between A and C?

Answer choices for question 1 A AB is positive and less than AC B BD bisects angle B C D must be the midpoint of AC D The longest side always has a midpoint at D
2

In the cutoff construction, D lies between A and C and AD equals AB. Which triangle has angle BDA as an exterior angle?

Answer choices for question 2 A Triangle ABD B Triangle ABC C Triangle BDC D No triangle in the construction
3

In triangle ABC, angle B is larger than angle C. An indirect proof considers the possibility $AC<AB$ . What contradiction follows?

Answer choices for question 3 A Angle A would be zero B Angle C would be larger than angle B C AC would equal BC D Angle B would be a right angle
4

Triangle ABC has angle measures
$m\angle A=(2x+10)^\circ.$ $m\angle B=(x+20)^\circ.$ $m\angle C=(150-3x)^\circ.$
For which real x do these form a triangle with BC strictly longest?

Answer choices for question 4 A $10<x<50$ B $28\le x<50$ C $28<x<60$ D $28<x<50$
5

An isosceles triangle ABC has AB equal to AC and vertex angle
$m\angle A=(2x+20)^\circ.$
Which value of x gives a valid triangle with base BC strictly longer than either equal side?

Answer choices for question 5 A 20 B 10 C 25 D 80
6

In triangle ABC, angle A is larger than angle B, and angles B and C are equal. Which pair of side comparisons follows?

Answer choices for question 6 A
$AC>BC$

$AC=AB$
B
$BC>AC$

$AC=AB$
C
$AB>BC$

$AB=BC$
D
$BC=AC$

$AC>AB$
7

H, Q, and R lie in that order on a line. P is outside the line and PH is perpendicular to it. Which strict comparison follows?

Answer choices for question 7 A $PR>PQ$ B $PQ>PR$ C $PH>PR$ D $PQ=PR$
8

P is outside a line, H is its perpendicular foot, and Q and R are different points on the line. Suppose
$PH=5.$ $PQ=4x-1.$ $PR=2x+7.$
Which interval satisfies exactly these two comparisons?
$PQ>PH.$ $PQ<PR.$

Answer choices for question 8 A $x>4$ B $x<\tfrac32$ C $\tfrac32\le x\le4$ D $\tfrac32<x<4$
9

In an existing triangle ABC, AB is 5 units, AC is 8 units, and angle B is 80 degrees. Which statement about angle C must be true?

Answer choices for question 9 A Its measure equals 50 degrees B Its measure is less than 80 degrees C Its measure is greater than 80 degrees D Its measure equals 80 degrees
10

For the converse theorem, suppose angle B is larger than angle C in triangle ABC. Which two alternatives must be ruled out before concluding $AC>AB$ ?

Answer choices for question 10 A AC equals AB, or AC is less than AB B AC equals BC, or AC is greater than BC C AB is positive, or AC is positive D AC is greater than AB, or AC equals AB
11

A triangle has an exterior angle of 70 degrees and a remote interior angle of 25 degrees. How does that exterior angle compare with its adjacent interior angle?

Answer choices for question 11 A The exterior angle is 40 degrees larger B They are equal C The exterior angle is 25 degrees larger D The exterior angle is 40 degrees smaller
12

Triangle ABC has $AB>AC$ . Which additional fact would force BC to be strictly longest?

Answer choices for question 12 A Angle B is larger than angle A B Angles A and B are equal C Angle A is larger than angle C D Angle A is smaller than angle B

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