Download 6.2 Similar Triangle Theorems

GEOMETRY
6.2 Similar Triangles or Not?
6.2 Similar Triangle Theorems
• Objectives
• Use constructions to explore similar triangle theorems
• Explore the Angle-Angle (AA~) Similarity Theorem
• Explore the Side-Side-Side (SSS~) Similarity Theorem
• Explore the Side-Angle-Side (SAS~) Similarity Theorem
• Key Terms
• Included Angle and Included Side
What do we need to set up to solve this problem?
A PROPORTION
Equal ratios that compare units
𝑙𝑏𝑠.
𝑝𝑒𝑜𝑝𝑙𝑒
35
𝑥
=
100 230
35 ∙ 230
𝑥=
100
𝑥 = 80.5 𝑙𝑏𝑠. 𝑜𝑓 𝑠𝑢𝑔𝑎𝑟
6.2 Similar Triangle Theorems
Skipping 6.1
Problem 1: Using Two Angles
• What does it mean for figures to be similar?
• Two polygons are similar if all corresponding angles are
congruent and the ratios of the measures of all
corresponding sides are equal. (Pg. 446)
• Pg. 452 Collaborate #1 (2 Minutes)
∆𝑅𝑆𝑇~∆𝑊𝑋𝑌
• Corresponding Angles
• ∠𝑅 ≅ ∠𝑊, ∠𝑆 ≅ ∠𝑋, ∠𝑇 ≅ ∠𝑌
• Corresponding Proportional Sides
•
𝑅𝑆
𝑊𝑋
𝑆𝑇
𝑅𝑇
= 𝑋𝑌 = 𝑊𝑌
Problem 1: Using Two Angles
• We only need two congruent corresponding angles to say
that two triangles are similar.
• We can prove this by using constructions but we are not
going to do those today
• Skip 2-4 Construction
Problem 1: Using Two Angles
• Angle-Angle Similarity Theorem (AA~)
• If two angles of one triangle are congruent to two angles of another
triangles, then the triangles are similar.
• Together #5
Problem 1: Using Two Angles
• Collaborate 6-7 (2 Minutes)
#6
No, the angles in the individual triangles are congruent, but
we don’t know anything about transferring between the
separate triangles.
Problem 1: Using Two Angles
#7
Yes, the angles in the individual triangles are congruent
and the vertex angles are congruent which allows us to
know that both sets of base angles are congruent.
Problem 2: Using Two and
Three Proportional Sides
• Skip 1-6 Construction
• Side-Side-Side Similarity Theorem (SSS~)
• If the corresponding sides of two triangles are proportional, then the
triangles are similar.
Problem 2: Using Two and
Three Proportional Sides
• Collaborate 7-8 (3 Minutes)
#7
If the lengths of the sides are known, then the measures of
the angles in the triangle are known. (We could find them
all: Later Lessons)
Problem 2: Using Two and
Three Proportional Sides
𝑈𝑉 33 3
=
=
𝑋𝑌 22 2
𝑉𝑊 24 3
=
=
𝑌𝑍
16 2
𝑈𝑊 36 3
=
=
𝑋𝑍
24 2
The triangles are similar because the ratios of the corresponding sides are equal
Problem 3: Using Two Proportional
Sides and an Angle
• Included Angle
• An angle formed by two consecutive sides of a figure.
• The angle between the two sides used.
• Included Side
• A line segment between two consecutive angles of a figure.
• The side between the two angles used.
• Skip 1-4 Construction
Problem 3: Using Two Proportional
Sides and an Angle
• Side-Angle-Side Similarity Theorem (SAS~)
• If two of the corresponding sides of two triangles are proportional
and the included angles are congruent, then the triangles are
similar.
Talk the Talk Pg. 460
Angle-Angle
Side-Side-Side
Side-Angle-Side
Reminder: Parallel Lines
Reminder: Reflexive Property
When 2 lines are ∥,
angles are either
congruent or
supplementary.
Congruent to itself
AA~
Ratio of 2 sets of sides =
SAS~
1
3
Formative Assessment
• Skills Practice 6.2
• Problem Set: Pg. 519-526 (1-20)
AA~
SSS~
SAS~