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Name ________________________
Geometry
Date ___________________
Schwimmer
Midterm Review Day 3- Triangles
1.
The measure of a base angle of an isosceles triangle is 13 more than 3 times the measure
of the vertex angle. How many degrees are in the vertex angle?
(1)
(2)
(3)
(4)
2.
11
22
33
42
x
7 x + 26 = 180
7 x = 154
x = 22
3x + 13
3x + 13
Which set of numbers could represent the lengths of the sides of an isosceles triangle?
(1)
(2)
(3)
(4)
{4, 5, 7}
{8, 8, 16}
{6, 6, 13}
{5, 5, 9}
In a triangle, the sum of any two sides must
be greater than the third.
In an isosceles triangle, two sides are the same length.
3.
A sign is 8 feet high and casts a 5-foot shadow while a nearby flagpole casts a 20-foot
shadow. How high is the flagpole?
8 x
=
5
20
(1) 32 feet
x
160 = 5 x
(2) 12.5 feet
8
(3) 2 feet
32 = x
(4) 10 feet
5
20
4.
The lengths of the sides of a right triangle may be
(1) 5,7, 8
(2) 7, 8, 12
(3) 7, 9, 11
(4) 8, 15, 17
Test in the Pythagorean Theorem!!
5.
Find the exact length of an altitude of an equilateral triangle whose side measures 10 cm.
(1)
(2)
(3)
(4)
10 2
10 3
5 2
5 3
30°
2x 10
x 3
90°
60°
x
6.
The perimeter of a square is 24. Find the length of the diagonal of the square.
6 2
6
x
45°
x 2
45°
7.
x 6
Find x in simplest radical form.
12 x
=
x 16
x 2 = 192
x = 64 3
x =8 3
8.
In right ∆ABC , altitude CD is drawn to hypotenuse AB. CD = 12 cm, and AD exceeds
BD by 7 cm. Find BD.
x + 7 12
=
12
x
2
x + 7 x = 144
12
x 2 + 7 x − 144 = 0
B
x
D
x+7
A
( x + 16 )( x − 9 ) = 0
x = −16 x = 9
BD = 9
9.
In the accompanying diagram of ∆ABC , D and E are midpoints. If DE = 6x – 8 and
CB = 8x + 4, find the length of CB.
2 ( 6 x − 8) = 8x + 4
12 x − 16 = 8 x + 4
4 x = 20
x=5
CB = 8(5) + 4
CB = 44
10.
In the accompanying diagram of isosceles triangle ABC, ∠ACB is the vertex angle,
CM ⊥ AB, and M is the midpoint of AB.
Which statement cannot be used to justify ∆ACM ≅ ∆BCM ?
(1)
(2)
(3)
(4)
11.
If the measures of the angles of a triangle are represented by 2x, 4x, and 6x, then the
triangle is
12 x = 180
(1)
(2)
(3)
(4)
12.
HL ≅ HL
AAS ≅ AAS
SSS ≅ SSS
AAA ≅ AAA
x = 15
2(15) = 30
right
obtuse
acute
equiangular
4(15) = 60
6(15) = 90
In the accompanying diagram, ABC , DBE , and EC are drawn and m∠ABD = 3 x.
What is the sum of m∠C and m∠E ?
(1)
(2)
(3)
(4)
13.
2x
3x
180 – x
180 – 3x
In the accompanying diagram of ∆ABC , a right angle is at C, AB = 8, and AC = 4.
What is the value of BC?
(1) 12
(3) 4 5
(2) 4 3
(4) 4
4 2 + x 2 = 82
16 + x 2 = 64
x 2 = 48
x = 16 3
x=4 3
14.
Determine if the measures 16, 30, and 34 can be the lengths of the sides of a right
triangle. Explain how you know.
16 2 + 30 2 = 34 2
256 + 900 = 1156
Yes, the lengths of these sides
satisfy the Pythagorean theorem.
1156 = 1156
15.
In right ∆ABC , altitude CD is drawn to hypotenuse AB. Use the accompanying
diagram to find AD.
x + 7 12
OOPS!
=
12
x
Repeat!
2
x + 7 x = 144
x 2 + 7 x − 144 = 0
( x + 16 )( x − 9 ) = 0
x = −16 x = 9
BD = 9
16.
Given: EA || DF , BE ⊥ AD, CF ⊥ AD, AC ≅ BD.
Prove: BE ≅ CF .
Statement
1. EA || DF
2. BE ⊥ AD
3. CF ⊥ AD
4. AC ≅ BD
5. BC ≅ BC
6. AB ≅ CD
7. ∠A ≅ ∠D
Reason
1. Given
2. Given
3. Given
4. Given
5. Reflexive
6. Subtraction
7. When two || lines are cut by a
transversal, alternate interior angles
are congruent.
8. ∠EBA and ∠FCD are right 8. ⊥ lines form right angles.
angles.
9. ∠EBA ≅ ∠FCD
10. ∆EAB ≅ ∆FDC
11. BE ≅ CF
9. All right angles are ≅ .
10. ASA ≅ ASA
11. CPCTC
17.
Complete the partial proof.
Given: Right ∆ABC with right angle C
BE ⊥ AB
Prove: AB × AD = AC × AE
Statement
Reason
1. Right ∆ABC with right angle C.
1.
Given
2. BE ⊥ AB
2.
Given
3.
∠ABE and ∠C are right angles.
3.
Perpendicular lines form right
angles.
4.
∠ABE ≅ ∠C
4.
All right angles are congruent.
5.
∠A ≅ ∠A
5.
Reflexive
6.
∆ABE ≅ ∆ACD
6.
AA Similarity
7.
AB AE
=
AC AD
7.
Corresponding sides of similar
triangles are in proportion.
8.
AB × AD = AC × AE
8.
The product of the means is equal to
the product of the extremes.