Download 9-5 Proving Triangles Similar Day 1

Name_________________________________
Geometry Period ________
Lesson 9-5 Notes
Date________
Lesson 9-5: Introduction to Similarity Proofs
Learning Goals: What are the minimum requirements to prove 2 triangles are similar? How can we use similarity to
prove proportions?
Proving Triangles are Similar
1) Show that the following two triangles are similar. Be sure to justify your work!
2) Justify that triangle ADE is similar to triangle ABC.
Re-draw diagram to see the two triangles clearly!
̅̅̅̅ and 𝑃𝑅
̅̅̅̅.
̅̅̅̅ intersect at T, and̅̅̅̅̅
Think! Which method could be used to prove ∆ PST ~ ∆RQT, given 𝑆𝑄
𝑃𝑆 ||𝑄𝑅
What can we infer using the given information? Mark the diagram!
Think – Pair – Share
Using the example from above, fill out the
box to the right:
*We can say that these two triangles are similar by the
____________ shortcut.
*Since they are similar we can set up the following proportion:
𝑋𝑌 𝑌𝑍
=
𝑃𝑅
Together!
After you have proven or shown that two triangles are similar, you will have another conclusion you can make.
If two triangles are similar then…
Using this Conclusion…
Example # 1:
Given triangle ABC, and triangle DEF. ∠ 𝐶𝐴𝐵 ≅ ∠𝐹𝐷𝐸 and ∠ 𝐶𝐵𝐴 ≅ ∠𝐹𝐸𝐷. Show that
I know ∆ 𝐴𝐵𝐶 is similar to ∆ 𝐷𝐸𝐹 because…
I know
because …
𝐷𝐸
𝐸𝐶
Example # 2: Given DE // AB. Prove that, 𝐵𝐴 = 𝐴𝐶 .
Think! What are the givens? What can we infer from the given? From the diagram?
MARK YOUR DIAGRAM!
Statement
Reason
1. DE // AB
1.
𝟐. ∠EDC ≅ ∠_________
2. Alternate interior angles are congruent when
lines are parallel.
𝟑. ∠DCE ≅ ∠BCA
3.
4. ∆𝑫𝑬𝑪 and ∆𝐵𝐴𝐶 are ________________.
4. AA (two corresponding angles are congruent)
5.
𝑫𝑪
𝑩𝑪
=
𝑫𝑬
𝑩𝑨
5.
Name_________________________________
Geometry Period ________
1)
In the diagram of
and
below,
Lesson 9-5 Practice
Date________
and
intersect at C, and
Redraw the triangles if it will help you! Which method can be used to show that
.
must be similar to
?
Fill in the following proofs:
2) Given
is an isosceles triangle with base AC and BD is perpendicular to AC,
prove.AB
BC
=
AD
.
CD
Statement
1.
Reason
is isosceles triangle with base AC and BD is
perpendicular to AC.
1.
𝟐. ∠BDA and ∠BDC are right angles.
2.
𝟑. ∠BDA ≅ ∠BDC
3. All right angles are congruent.
𝟒.
𝟒. ∠A ≅ ∠𝑩𝑪𝑨
5. ∆𝑫𝑬𝑪 and ∆𝑩𝑨𝑪 are ________________.
5. AA (two corresponding angles are congruent)
6. Prove AB * DC = AD * BC.
6.
3) Given Trapezoid ABCD:
Prove that ∆BCE ~ ∆DEA.
Name_________________________________
Geometry Period ________
Don’t forget to fill in your road map for today’s lesson!
Directions: Answer all the questions on this assignment. Show all work.
1) a) Name the similar triangles.
b) Justify that these triangles are similar. (Hint: what shortcut could we use to prove this?
c) Explain why
̅̅̅̅
𝑀𝐿
̅̅̅̅
𝑇𝑈
=
̅̅̅̅
𝐿𝑉
𝑈𝑉
2) ∆ACD is a right triangle with a right angle at c. AB ⊥ BE, which gives us a right angle at B.
a) Sketch ∆ ABE and ∆ACD separately.
b) Justify that ∆ ABE and ∆ACD are similar.
3) Given that ABCD is parallelogram.
a) Justify ∆𝐵𝐹𝐶 ~ ∆𝐸𝐹𝐷 .
b) Justify why
𝐵𝐶
𝐷𝐸
=
𝐵𝐹
𝐹𝐸
.
9-5 HW
Date________
4) Which triangles are always similar? Explain your choice.
(a) Equilateral triangles
(b) Isosceles triangles
Explanation:
(c) Right Triangles
(d) Scalene Triangles
5) Given: <STH and <FGH are congruent.
Prove: Triangle STH and triangle FGH are similar.
Statement
Reason
1. ∠STH and ∠FGH are congruent.
1.
𝟐. ∠____ ≅ ∠____
2. Vertical angles are congruent
3. ∆STH and ∆FGH are ________________.
3. AA (two corresponding angles
are congruent)
Mixed Review!
6) The lengths of two corresponding sides of two similar triangles are 18 inches and 12 inches. If an altitude of the
smaller triangle has a length of 6 inches, find the length of the corresponding altitude of the larger triangle.
7) Using the graph to the right:
a) Calculate the scale factor that dilates the preimage to the image.
b) What is the center of dilation?