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Angle Bisectors and the Incenter: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

Q lies outside angle UOV and is equally far from its two supporting side lines. What may be concluded from the interior-point converse?

Answer choices for question 1 A OQ must be the internal bisector B The two distances must be zero C The two supporting lines must be parallel D It does not identify OQ as the internal bisector because Q is exterior
2

Point P has equal distances to the full side lines of an angle, but its foot on one line lies behind the vertex. Why must line distance be distinguished from distance to the side ray?

Answer choices for question 2 A Every point on a line also lies on every ray of it B The perpendicular foot is allowed on the full line but is not on that forward ray C A perpendicular distance can be negative D The theorem compares only distances to endpoints
3

P is strictly inside triangle ABC and its perpendicular distances to all three side lines are equal. Which argument proves P is the incenter?

Answer choices for question 3 A Apply the perpendicular-bisector converse to A and B B Equal distances to side lines force the sides to be equal C Apply the interior-point converse at A and B, then use their unique intersection D Equal distances force P to lie on side AB
4

P is strictly inside triangle ABC and lies on the internal bisector at A. Which additional condition is sufficient to make P the incenter?

Answer choices for question 4 A Its perpendicular distances to AB and BC are equal B PA equals PB C AB equals AC D P has a known distance to A
5

Why do two different internal angle-bisector lines of a nondegenerate triangle determine at most one incenter?

Answer choices for question 5 A All lines have exactly one intersection B Two distinct lines have at most one common point C Every point inside a triangle is on a bisector D The triangle must have equal sides
6

A circle has center I and radius IF, where IF is perpendicular to line AB at F. Why can no other point T on line AB lie on that circle?

Answer choices for question 6 A IT is always shorter than IF B All points of AB are equally far from I C The radius IF is parallel to AB D IT is the hypotenuse of right triangle IFT and is longer than IF
7

U and V are distinct first-arc points. Two later circles centered at U and V use equal radius k, where k is less than half of UV. What happens?

Answer choices for question 7 A They do not intersect, so no point X is produced B They produce a unique internal-bisector point C They always meet at O D They make OU and OV unequal
8

A circle centered at an interior point P of a nondegenerate triangle ABC lies inside the triangle and touches AB and AC. A circle with the same center is drawn to touch AB and BC while staying inside the triangle. What must follow?

Answer choices for question 8 A The circles have different radii, and P lies only on the bisector at A B The circles coincide, and P must be the circumcenter C The circles coincide, and P is the incenter D P is the incenter, but the circles can have different radii
9

I is the incenter of triangle ABC and $\angle AIB=116^\circ$ . What is $\angle C$ ?

Answer choices for question 9 A $32^\circ$ B $64^\circ$ C $52^\circ$ D $128^\circ$
10

In the converse proof, an interior P has a perpendicular foot at the vertex O on one side line. Why cannot its two perpendicular distances be equal?

Answer choices for question 10 A That distance is OP, while the other is a leg shorter than hypotenuse OP B A perpendicular foot is never a vertex for any point C OP must be zero for every interior point D The other side line must be parallel to OP
11

A point inside triangle ABC is equally far from AB and AC but is closer to BC. Which conclusion is justified?

Answer choices for question 11 A It is the incenter because two distances agree B It is the circumcenter C It lies on all three internal bisectors D It lies on the internal bisector at A but is not the incenter
12

What supplies the geometric reason that one compass setting reaches all three sides in the incircle construction?

Answer choices for question 12 A Equal distances from I to the vertices B The equal perpendicular distances established by the bisector theorem and converse C The opposite-side ratio theorem by itself D The fact that every triangle has a longest side

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