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Chapter 6: Similar Triangles
Topic 5: Not – So – Formal Similar Triangle Proofs
Definition of Similar Triangles –
Two triangles are similar if and only if the corresponding sides are in _________________________
and the corresponding angles are _________________________.
If: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹
Sides:
̅̅̅̅
𝐴𝐵
̅̅̅̅
𝐷𝐸
=
̅̅̅̅
𝐵𝐶
̅̅̅̅
𝐸𝐹
=
̅̅̅̅
𝐴𝐶
̅̅̅̅
𝐷𝐹
<𝐴≅<𝐷
<𝐵 ≅<𝐸
< 𝐶 ≅< 𝐹
Angles:
Methods to prove Triangles are Similar –
1) 𝑨𝑨 ~ 𝑨𝑨
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
If:
<𝐴 ≅<𝐷
<𝐵 ≅<𝐸
Then: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹
2) 𝑺𝑺𝑺 ~ 𝑺𝑺𝑺
If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
If:
̅̅̅̅
𝐴𝐵
̅̅̅̅
𝐷𝐸
=
̅̅̅̅
𝐵𝐶
̅̅̅̅
𝐸𝐹
=
̅̅̅̅
𝐴𝐶
̅̅̅̅
𝐷𝐹
Then: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹
3) 𝑺𝑨𝑺 ~ 𝑺𝑨𝑺
If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the
sides including these angles are in proportion, the triangle are similar.
If:
̅̅̅̅
𝐴𝐵
̅̅̅̅
𝐷𝐸
̅̅̅̅
𝐴𝐶
= ̅̅̅̅
𝐷𝐹
<𝐴 ≅<𝐷
Then: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹
Practice:
1) How is ∆𝐴𝐵𝐸 ~ ∆𝐷𝐶𝐸?
2) Prove ∆𝐴𝐵𝐶 ~ ∆𝑃𝑄𝑅 using the given information.
Show a proof of why or why not.
3) Determine if ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹?
4) Prove ∆𝐸𝐴𝐷 ~ ∆𝐶𝐴𝐵 using the given information.
Show a proof of why or why not.
5) How is ∆𝐸𝐷𝐶 ~ ∆𝐴𝐵𝐶?
̅̅̅̅
𝐴𝐵 = 10
< 𝐴 = 40
̅̅̅̅ = 15
𝐴𝐶
̅̅̅̅
𝐷𝐸 = 20
< 𝐷 = 40
̅̅̅̅ = 30
𝐷𝐹
6) Given:
Prove: ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹
7) If ∆𝑅𝑆𝑇 ~ ∆𝐴𝐵𝐶, 𝑚 < 𝐴 = 𝑥 2 − 8𝑥, 𝑚 < 𝐶 = 4𝑥 − 5, and 𝑚 < 𝑅 = 5𝑥 + 30, find 𝑚 < 𝐶.
_____ 8) In triangles ABC and DEF, AB = 4, AC = 5, DE = 8, DF = 10 and <A ≅ <D. Which method could be used
to prove ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹?
(1) AA
(2) SAS
(3) SSS
(4) ASA
_____ 9) Scalene triangle ABC is similar to triangle DEF. Which statement is false?
(1) 𝐴𝐵 ∶ 𝐵𝐶 = 𝐷𝐸 ∶ 𝐸𝐹
(2) 𝐴𝐶 ∶ 𝐷𝐹 = 𝐵𝐶 ∶ 𝐸𝐹
(3) < 𝐴𝐶𝐵 ≅< 𝐷𝐹𝐸
(4) < 𝐴𝐵𝐶 ≅ < 𝐸𝐷𝐹
_____ 10) In the diagram, ∆𝐴𝐵𝐶 ~ ∆𝑅𝑆𝑇. Which statement is not true?
𝐴𝐵
𝐵𝐶
(1) < 𝐴 ≅< 𝑅
(2)
=
𝑅𝑆
(3)
𝐴𝐵
𝐵𝐶
=
𝑆𝑇
𝑅𝑆
_____ 11) In ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹,
(1) 𝐴𝐶 = 𝐷𝐹
(3) < 𝐴𝐶𝐵 ≅ < 𝐷𝐹𝐸
(4)
𝐴𝐶
𝐷𝐹
=
𝐶𝐵
.
𝐹𝐸
𝑆𝑇
𝐴𝐵+𝐵𝐶+𝐴𝐶
𝑅𝑆+𝑆𝑇+𝑅𝑇
=
𝐴𝐵
𝑅𝑆
Which additional information would prove ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹?
(2) 𝐶𝐵 = 𝐹𝐸
(4) < 𝐵𝐴𝐶 ≅ < 𝐸𝐷𝐹
Name ______________________________________________
Date ____________________
Not – So – Formal Similar Triangle Proofs HW
1) Find the scale factor that proves ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹. Then show that this scale factor holds true for every side
of the dilated figure.
2) How is ∆𝐴𝐵𝐶 ~ ∆𝐺𝐻𝐼?
3) Determine if ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹 or not. Show a proof of why or why not.
4) Determine if ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹?
5) Is ~ 𝛥𝐷𝐸𝐹? Prove why or why not.
6. Given: ̅̅̅̅̅
𝑃𝑅 = 15
𝑚 < 𝑃 = 70°
̅̅̅̅ = 21
𝑃𝑄
̅̅̅̅
𝑋𝑍 = 5
𝑚 < 𝑋 = 70°
̅̅̅̅ = 7
𝑋𝑌
Prove: 𝛥𝑃𝑄𝑅 ~ 𝛥𝑋𝑌𝑍
7) In the diagram, ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹, DE = 4, AB = x, AC = x + 2, and DF = x + 6. Determine the length of AC.
Review Section:
_____8) ∆𝐴𝐵𝐶 is shown in the diagram. If DE joins the midpoints of ADC and AEB,
which statement is not true?
1
(1) DE = 2 𝐶𝐵
(2) DE // CB
(3)
𝐴𝐷
𝐷𝐶
=
𝐷𝐸
𝐶𝐵
(4) ∆𝐴𝐵𝐶 ~ ∆𝐴𝐸𝐷
_____9) Given that ABCD is a parallelogram, a student wrote the proof
below to show that a pair of opposite angles are congruent.
What is the reason justifying that < 𝐵 ≅ < 𝐷?
(1) Opposite angles in a quadrilateral are congruent
(2) Parallel lines have congruent corresponding angles
(3) Corresponding parts of congruent triangles are congruent
(4) Alternate interior angles in congruent triangles are congruent
_____10) In ∆𝐴𝐵𝐶, DE // BC. If AB = 10, AD = 8, and AE = 12,
What is the length of EC?
(1) 6
(2) 2
(3) 3
(4) 15
_____11) What is the slope of a line perpendicular to the line whose equation is 20𝑥 − 2𝑦 = 6?
1
(1) −10
(2) − 10
(3) 10
(4)
1
10