Theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
a two-column proof of the theorem is shown, but the proof is incomplete.
a triangle with vertices a is at 6, 8. b is at 2, 2. c is at 8, 4. segment de with point d on side ab and point e is on side bc.
statement reason
the coordinates of point d are (4, 5) and coordinates of point e are (5, 3) by the midpoint formula
length of segment de is square root of 5 and length of segment ac is 2 multiplied by the square root of 5 by the distance formula
segment de is half the length of segment ac by substitution
slope of segment de is −2 and slope of segment ac is −2
segment de is parallel to segment ac slopes of parallel lines are equal
which of the following completes the proof?
by definition of congruence
addition property of equality
by construction
by the slope formula
question 8(multiple choice worth 6 points)
(05.03 lc)
use the figure below to answer the question that follows:
intersecting triangles ace and bdf. they intersect at points g, h, i, and j.
what must be given to prove that δcjg ~ δcea?
∠cjg ≅ ∠cae
segment gj is parallel to segment ae
∠chi ≅ ∠cgj
segment hi is parallel to segment ae
question 9(multiple choice worth 6 points)
(05.03 lc)
if segment ln is congruent to segment np and ∠1 ≅ ∠2, prove that ∠nlo ≅ ∠npm:
overlapping triangles lno and pnm. the triangles intersect at point q on segment lo of triangle lno and segment mp of triangle pnm.
hector wrote the following proof for his geometry homework for the given problem:
statements reasons
segment ln is congruent to segment np given
∠1 ≅ ∠2 given
∠n ≅ ∠n reflexive property
δlno ≅ δpnm angle-angle-side postulate
∠nlo ≅ ∠npm
which of the following completes hector's proof?
angle addition postulate
converse of corresponding angles postulate
corresponding parts of congruent triangles are congruent
triangle proportionality theorem
question 10(multiple choice worth 6 points)
(05.03 mc)
the figure below shows a quadrilateral abcd with diagonal bd bisecting angle adc:
a quadrilateral abcd is drawn with bd as a diagonal. bd bisects angle adc and forms two triangles abd and cbd. ad and cd are equal in length.
which equation is true?
angle dcb = angle adb
angle dab = angle dcb
angle abd = angle bcd
angle abd = angle cdb