Angle Relationships
Name the angle relationship and solve for x.
vertical angles
x = 10
Which is NOT a valid congruence theorem?
A. Angle-Angle-Side
B. Angle-Angle-Angle
C. Side-Side-Angle
D. Side-Side-Side
B (Angle-Angle-Angle) and C (Side-Side-Angle)
If you know that two triangles are similar, what do you know about their angles and side lengths?
their angles are congruent and their side lengths are proportional
proofs start with what type of statement?
the given(s)
Angles that have the same size are called...
congruent
Justifying each step, find the measure of ∠x.
1. ∠BAC (the third angle in the triangle) is 180 - 68 - 70 = 42 degrees, by the triangle sum theorem
2. ∠BAC and ∠x are supplementary and thus sum to 180, so 180 - 42 = 138
∠x = 138
What two theorems do you need in order to prove that these triangles are congruent?
triangle sum theorem and ASA≅
Given that △ABC is similar to △DEF, write three ratios in a proportion to describe their side lengths.
AB / DE = BC / EF = CA / FD
if this is a statement in a proof, what could the reason be? (list 2 possibilities)
∠A + ∠B + ∠C = 180
supplementary
triangle sum theorem
Angles that sum to 90 are called _____________, while angles that sum to 180 are called _____________
complementary, supplementary
Find the measures of angles f and e.
∠f = 58
∠e = 26
Write a congruence statement and state the congruence theorem used.
△FGH ≅ △IJH
AAS≅
Given △ABC and △DEF, if ∠A = 20, ∠D = 40, ∠B = 30, and ∠E = 60, are the triangles similar? Justify why or why not.
no, similarity means the angles are congruent (not scaled)
provide possible reasons for each of the following statements:
∠A = 50
∠A + ∠B = 90
50 + ∠B = 90
∠B = 40
∠A = 50 (given)
∠A + ∠B = 90 (complementary)
50 + ∠B = 90 (substitution)
∠B = 40 (subtraction)
vertical angles and corresponding angles:
name 1 similarity and 1 difference
similarity: both vertical and corresponding angles are congruent
difference: vertical angles are reflected across the vertex, while corresponding angles are slid along the transversal
Find the measures of angles a, b, c, and d, and explain using correct angle relationships vocabulary.
∠a = 102
∠b = 78
∠c = 58
∠d = 122
Prove that the triangles are congruent.
Hint: last step should be the congruence statement.
1. side UV ≅ side WV (given)
2. side TU ≅ side TW (given)
3. side TV ≅ side TV (reflexive)
4. △TUV ≅ △TWV (SSS≅)
Given △ABC and △DEF, if AB = 6, BC = 9, DE = 8, and EF = 12, are the triangles similar? Explain.
they may, or may not, be similar
the given side lengths are proportional
but we need the included angle or the third side to prove similarity
Two right triangles share a common side AB with vertex B, and ∠ABC is congruent to ∠ABD. Prove that the triangles are congruent.
1) ∠BAC ≅ ∠BAD (given - both right angles)
2) ∠ABC ≅ ∠ABD (given)
3) AB = AB (reflexive property)
4) △ABC ≅ △ABD
draw an example of, or clearly describe, each of the following:
alternate interior angles
alternate exterior angles
same side interior angles
same side exterior angles
alternate interior: opposite sides of the transversal, between the parallel lines
alternate exterior: opposite sides of the transversal, outside the parallel lines
same side interior: same side of the transversal, between the parallel lines
same side exterior angles: same side of the transversal, outside the parallel lines
Find the measure of ∠h and explain all reasoning.
∠h = 17
If △COW ≅ △PIG, CO = 25, CW = 18, IG = 23, and PG = 7x - 17, find x and PG.
x = 5
PG = 18
Given △ABC and △DEF, if ∠A = 42, ∠C = 81, ∠D = 81, and ∠F = 57, are the triangles similar? Explain.
yes:
∠A corresponds to ∠E (both 42)
∠C corresponds to ∠D (both 81)
∠B corresponds to ∠F (both 57)
for a total of 180 degrees (triangle sum theorem)
AAA~
Given △ABC, D is a point on AB and E is a point on AC such that DE is parallel to BC. Prove that △ADE is similar to △ABC.
∠ADE ≅ ∠ABC (corresponding)
∠AED ≅ ∠ACB (corresponding)
∠DAE ≅ ∠BAC (reflexive)
△ADE ~ △ABC (AA~)
any side or angle is congruent to itself
useful in congruence/similarity theorems when triangles have an overlapping side or angle