The Angle-Bisector Ratio Theorem: Practice Level 1 | Geometry

Level 1 · Foundational 0 / 12 answered

Question 1 of 12

1

In nondegenerate triangle ABC, AD internally bisects angle BAC and D is strictly between B,C. Which equality follows?

Answer choices for question 1 A $\frac{BD}{BC}=\frac{AB}{AC}$ B $\frac{DC}{BD}=\frac{AB}{AC}$ C $\frac{BD}{DC}=\frac{AB}{AC}$ D $\frac{BD}{DC}=\frac{AD}{AC}$
2

What does it mean for AD to be the internal bisector of angle BAC in a nondegenerate triangle?

Answer choices for question 2 A D lies between B,C and angles BAD,DAC are equal B D is always the midpoint of BC C AD must be perpendicular to BC D AB and AC must be equal
3

AD is an internal bisector and $AB:AC=5:11.$ What is BD:DC?

Answer choices for question 3 A $11:5$ B $5:16$ C $1:1$ D $5:11$
4

AD is an internal bisector and $AB=AC.$ What follows for D?

Answer choices for question 4 A D lies beyond C B D is the midpoint of BC C BD is twice DC D AD has the same length as BC
5

An internal bisector divides an angle measuring $74^\circ$ . What is each half-angle measure?

Answer choices for question 5 A $74^\circ$ B $148^\circ$ C $37^\circ$ D $16^\circ$
6

Which lengths does this ratio theorem compare?

Answer choices for question 6 A Perpendicular distances from a point to the angle lines B Parts along the opposite side and the adjacent side lengths C The lengths of the two half-angle arcs D Only the length of the bisector
7

In the parallel proof, which line is drawn through C?

Answer choices for question 7 A A line parallel to AD B A line perpendicular to AD C A line parallel to BC D The angle bisector at C
8

In that auxiliary construction, which point order on the extended side is correct?

Answer choices for question 8 A $B,E,A$ B $A,B,E$ C $A,E,B$ D $B,A,E$
9

Why does the proof establish $AE=AC$ ?

Answer choices for question 9 A Parallel lines always have equal lengths B Equal angles at E and C make triangle AEC isosceles C The whole triangle ABC must be equilateral D AD must bisect BC
10

D is strictly between B,C and $BD/DC=AB/AC$ in a nondegenerate triangle. What does the converse conclude?

Answer choices for question 10 A D is the midpoint of BC in every case B AB equals AC in every case C AD is parallel to BC D AD internally bisects angle BAC
11

If AD internally bisects angle BAC, which reciprocal form is valid?

Answer choices for question 11 A $\frac{DC}{BD}=\frac{AB}{AC}$ B $\frac{BC}{BD}=\frac{AC}{AB}$ C $\frac{DC}{BD}=\frac{AC}{AB}$ D $\frac{AD}{DC}=\frac{AC}{AB}$
12

Why is AB/AC not automatically a dilation scale factor here?

Answer choices for question 12 A It compares two sides of one triangle B Every side ratio is a scale factor C Only ratios greater than 1 can be scale factors D The theorem uses negative lengths