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G.CO.10 ASSESSMENT – PATTERSON 1 MULTIPLE CHOICE 1. If isosceles ΔABC has its vertex angle at ∠C, then A) AB ≅ AC B) BA ≅ BC C) CA ≅ CB D) ∠C ≅ ∠B C) m∠A = 40° D) m∠A = 24° 2. ΔABC has m∠C = 24° and m∠B = 40°, then A) m∠A = 26° B) m∠A = 116° 3. ΔABC has m∠C = 45° and m∠A = 45°, then the correct classification for the triangle is: A) Isosceles B) Acute C) Right D) A & C 4. ΔABC has m∠C = 42° and m∠A = 80°, then the correct classification for the triangle is: A) Acute B) Right C) Obtuse D) Isosceles 5. If isosceles ΔABC has a base angle, ∠B, with a measure of 40°, then ΔABC is A) an acute Δ B) a right Δ C) an obtuse Δ D) a scalene Δ 6. If isosceles ΔABC has a base angle, ∠B, with a measure of 50°, then ΔABC is A) an acute Δ B) a right Δ C) an obtuse Δ D) a scalene Δ 7. If isosceles ΔABC has a base angle, ∠B, with a measure of 45°, then ΔABC is A) an acute Δ B) a right Δ C) an obtuse Δ D) a scalene Δ A
8. DE is the mid-­‐segment of ΔABC. Which of the following is a false statement? A) DE = 2BC B) AD = EC C) 2DE = BC D) ∠B ≅ ∠C E
C
D
B
9. DE is the mid-­‐segment of ΔABC. If DE = 7.4 cm, then A) BC = 14.8 cm B) AD = 7.4 cm C) AB = 14.8 cm D) m∠B = 14.8° A
E
C
D
B
G.CO.10 ASSESSMENT – PATTERSON 10. DE is the mid-­‐segment of ΔABC. Which of the following is true? A) AD = 2AB B) BC = 2DE C) AB = AC D) ∠B ≅ ∠C 2 A
E
C
D
B
11. DE is the mid-­‐segment of ΔABC and m∠A = 90°. If AD = 3 cm and AE = 4 cm, then: A) BC = 14 cm B) BC = 10 cm C) BC = 8 cm D) BC = 6 cm E
C
A
D
B
Answers: 1. C 2. B 9. A 10. B 3. D 4. A 5. C 6. A 7. B 8. A 1. ΔABC has m∠A = 16° and m∠C = 57°, then the triangle is an obtuse triangle. T or F 2. ΔABC has m∠A = 44° and m∠C = 46°, then the triangle is a right triangle. T or F 3. ΔABC has m∠A = 30° and m∠C = 30°, then AB ≅ AC . T or F 4. ΔABC has m∠A = 63° and m∠C = 63°, then the triangle is an acute triangle. T or F 5. ΔABC has m∠C = 71° and m∠B = 71°, then AB ≅ AC . T or F T or F 7. If isosceles ΔABC has a base angle, ∠B, of 59°, then vertex ∠A = 61° T or F 8. If isosceles ΔABC has a base angle, ∠B, of 59°, then base ∠C = 59° T or F 9. If a base angle of an isosceles is 70°, then the vertex angle is acute. T or F 10. An equilateral triangle is not an isosceles triangle. T or F 11. An isosceles triangle cannot have a right angle because it would exceed 180°. T or F 12. An acute triangle has exactly two acute angles. T or F 11. B TRUE/FALSE 6. If isosceles ΔABC has a vertex angle, ∠B, of 83°, then m∠A = 49° G.CO.10 ASSESSMENT – PATTERSON 3 13. The two acute angles in a right triangle are always complementary. T or F 14. An equilateral triangle is defined by three equal sides. T or F 15. All equilateral triangles are isosceles triangles. T or F 5. T 6. F 7. F Answers: 1. T 2. T 9. T 10. F 3. F 4. T 11. F 12. F 13. T 14. T 8. T 15. T SHORT ANSWER 1. Jennifer has seen an informal proof that the three angles of a triangle sum to 180°. She draws a triangle on a piece of paper and then cuts it out. She then tears off the three corners (angles) of the triangle. What does she do next? How does this help to support that the sum of a triangle is 180°? Answer: She would place the three angles together so that the vertices all touch and they adjacent. When this happens the three angles form a straight line. A straight angle is equal to 180°… thus the sum of the angles is 180°. 2. Determine the value(s) of the variables. a) x = ________ b) x = ________ 98°
2x
4x
63°
x
52°
71°
Answer: a) x = 30 c) x = ________ b) x = 11.5 75°
c) x = 7.5 3. Determine the value(s) of the variables. a) x = ________ 14°
b) x = ________ 3x
x
12°
4x
2x
Answers: a) x = 154 c) x = ________ b) x = 20 c) x = 23 60°
3x
2x + 5
G.CO.10 ASSESSMENT – PATTERSON 4 4. Determine the value(s) of the variables. a) x = ________ b) x = ________ 126°
x
41°
89°
35°
x
165°
x
Answers: a) x = 36 c) x = ________ b) x = 76 c) x = 76 5 Determine the value(s) of the variables. a) x = ________ b) x = ________ c) x = ________ x
32°
x
53°
x
x
x
Answers: a) x = 74 b) x = 74 c) x = 60 6 Determine the value(s) of the variables. a) x = ________ b) x = ________ c) x = ________ 5x - 7
x
94°
33°
17°
d) x = ________ 37°
x + 25
x
e) x = ________ f) x = ________ 87°
33°
70°
x
60°
Answers: 70°
x
a) x = 55.5 b) x = 1 2 c) x = 106.5 x
57°
d) x = 55 e) x = 107 f)x = 38 G.CO.10 ASSESSMENT – PATTERSON 5 7 Determine the missing information a) m∠1 = ________ b) x = ________ m∠2 = ______ 2
58°
22°
65°
1
39°
x
68°
79°
Answers: a) x = 68 x = 51 b) 90° 8. If the legs of an isosceles triangle are 6x – 6 cm and x + 9 cm long and the base is 2x + 10 cm. Find the length of the base. Answer: 6x – 6 = x + 9 x = 3 so the base is 2(3) + 10 = 16 cm 9. Complete the following:
a) If ΔDOG is isosceles and OG ≅ OD , then the base angles are ________ and _______.
b) If two angles of a triangle are 48 & 67, then the 3rd angle is ________.
c) If one remote angle is 55° and the exterior angle is 135°. What is the other remote angle? _________
d) What is the full name of the ΔABC with the following given information
-- m∠A = 60°, m∠B = 80°, m∠40° _______________________________________________
-- AB = AC & m∠A = 130°
________________________________________________
-- AB > BC > CA
________________________________________________
Answers: a) ∠G & ∠D b) 65° c) 80° d) Scalene Acute, Obtuse Isosceles, Scalene 10. If the base of the isosceles triangle is called a hypotenuse, what is the full name of the triangle? Answer: Isosceles Right Triangle 11. Explain why a triangle cannot have two obtuse angles in it. Answer: An obtuse angle is defined to be greater than 90° and if you double that you exceed the angle sum of the interior angles of a triangle (180°). Thus this can’t happen!! G.CO.10 ASSESSMENT – PATTERSON 6 12. Draw ΔALT using the following information. Label your diagram completely so we know if sides or angles are congruent or different. If it is not possible to draw such a triangle, simply write NOT POSSIBLE. a) A Scalene Right Triangle, m∠T = 90° (ΔALT) b) A Right Equilateral (ΔALT) c) AL ≅ AT , m∠A > 90° (ΔALT) d) m∠A = m∠L = m∠T (ΔALT) Answers: a) b) IMPOSSIBLE L
T
A
c) d) A
T
L
A
o
L
T
13. The measure of a base angle of an isosceles triangle is 52 degrees. What is the measure of the vertex angle? Answer: 76° 14. If ΔMAE ≅ ΔGTD and ME is the base of the isosceles triangle. What are the legs of ΔGTD? _______ and ________ Answer: TD & TG are the legs LONG ANSWER 1. Prove that the sum of the interior angles of a triangle is 180° in two DIFFERENT WAYS. a) Classic Approach An informal proof that is often used is the process of having our students create a triangle on a piece of paper, naming the three angles A, B, and C and then cutting out the triangle. When the triangle is cut out, the student should rip off the three angles, placing them together, vertex to vertex. They will see that the three angles form a straight line. Therefore, the sum of the three interior angles of a triangle is 180°. G.CO.10 ASSESSMENT – PATTERSON 7 b) Informal Approach B
Given: ΔABC Prove: m∠1 + m∠2 + m∠3 = 180°
2
1
Construct an auxiliary line parallel to AC through B. STATEMENT REASON Given (Auxiliary Line) AC || BD m∠4 + m∠2 + m∠5 = 180° m∠1 = m∠4 m∠3 = m∠5 m∠1 + m∠2 + m∠3 = 180° 3
A
C
D
Angles of a Straight Angle If P , Alternate Interior ∠’s ≅ If P , Alternate Interior ∠’s ≅ Substitution Property (Twice) c) Transformational Approach Given: ΔABC Prove: m∠1 + m∠2 + m∠3 = 180° B
2
A
1
uuur
Translate ΔABC by vector AB to form straight ∠ABB’ along the vector. The isometric properties of translation preserve angles, thus m∠1 = m∠B’BC’. Since ∠ABB’ is a straight angle, we know that m∠2 + m∠CBC’ + m∠B’BC’ = 180°. Translations also preserve parallelism, therefore ensuring that AC P BC ' . Since AC P BC ' m∠3 = m∠CBC’ because alternate interior angles are congruent. By making two substitutions into the straight angle relationship of m∠2 + m∠CBC’ + m∠B’BC’ = 180° we arrive at the proof that m∠1 + m∠2 + m∠3 = 180°. B'
1
B =A'
2
A
1
C
3
C'
3
3
C
d) Transformational Approach B
Given: ΔABC 2
Prove: m∠1 + m∠2 + m∠3 = 180° 1
Rotate ΔABC 180° about the midpoint of BC forming image ΔDCB with congruent corresponding angles (CPCTC). Rotate ΔDCB about the midpoint of BD forming image ΔBED with congruent corresponding angles (CPCTC). Because of the congruent alternate interior A
3
C
G.CO.10 ASSESSMENT – PATTERSON angles formed (∠ABC ≅ ∠DCB and ∠EBD ≅ ∠CDB) , 8 E
AB P DC and DC P BE , respectively. In addition, 2
because there is only one line parallel to DC through 1
B
point B, ∠ABE is a straight ∠ formed by ∠1, ∠2 and ∠3. Thus, m∠1 + m∠2 + m∠3 = 180°. 3
2
A
D
3
1
1
3
2
C
2. Prove that the sum of the interior angles of a triangle is 180° in a transformational approach. Answers: Provided in Long Answer Q1 3. Prove that the base angles of an isosceles triangle are congruent using two different ways. a) A traditional way Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C B
Construct an auxiliary line that is the angle bisector of ∠B. STATEMENT REASON Given ΔABC is an isosceles triangle. BD is the angle bisector of ∠B Given (Auxiliary Line) Definition of an Isosceles Δ BA ≅ BC ∠ABD ≅ ∠CBD BD ≅ BD ΔABD ≅ ΔCBD ∠A ≅ ∠C
Definition of an Angle Bisector Reflexive Prop. (Common Side) SAS CPCTC (Corresponding Parts of Congruent Triangles are Congruent) A
D
C
G.CO.10 ASSESSMENT – PATTERSON 9 b) A transformational way Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C Construct an auxiliary line; BD such that BD is the perpendicular B
bisector of AC Established in G.CO.3, an isosceles triangle has reflectional symmetry about the perpendicular bisector of its base. Thus ∠A ≅ ∠C because ∠A reflects onto ∠C. A
D
C
D
C
c) A formal method Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C Construct an auxiliary line that is a perpendicular bisector of AC . By the definition of isosceles triangle, AB ≅ CB. Because point B is equidistant to points A and C, B is on the perpendicular bisector of AC . A reflection over the perpendicular bisector would map A onto C, B onto B, and D onto D. Thus the isometric properties of a reflection then give us ΔABD ≅ ΔCBD. Therefore, ∠A ≅ ∠C by corresponding parts of congruent triangles are congruent. B
A
4. Prove that the base angles of an isosceles triangle are congruent using a transformational approach. a) Informal Method Prove that the base angles of an isosceles triangle are congruent. Create an isosceles triangle by using your compass to construct a circle. Then draw two radii (all radii of the same circle are congruent) and connect the endpoints with a segment. Label the radii, AB and CB . Then fold the paper until point A maps to point C. Crease the paper. Notice that when you do this ∠A ≅ ∠C. Therefore, the base angles of an isosceles triangle are congruent. B
A
D
C
G.CO.10 ASSESSMENT – PATTERSON 10 b) A formal method Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C Construct an auxiliary line; BD such that BD is the perpendicular B
bisector of AC Established in G.CO.3, an isosceles triangle has reflectional symmetry about the perpendicular bisector of its base. Thus ∠A ≅ ∠C because ∠A reflects onto ∠C. A
D
C
D
C
5. Prove the following Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C Construct an auxiliary line that is a perpendicular bisector of AC . (B is on the perpendicular bisector of AC because AB ≅ BC ) STATEMENT REASON Given ΔABC is an Isosceles. BD is the ⊥ bisector of ∠B Given (Auxiliary Line) Answer: Provide in Long Answer 3c B
A
G.CO.10 ASSESSMENT – PATTERSON 11 6. Prove the following Given: ΔABC is an isosceles triangle, with base AC . Prove: ∠A ≅ ∠C Construct an auxiliary line that is the angle bisector of ∠B. STATEMENT REASON Given ΔABC is an isosceles triangle. BD is the angle bisector of ∠B Given (Auxiliary Line) B
C
D
A
Answer: Provide in Long Answer 3a 7. An exterior angle is formed between a side and the extension of a side. It will always be a linear pair with an internal angle. In the diagram below, ∠4 is the exterior angle. The exterior angle theorem states that the EXTERNAL ANGLE IS EQUAL TO THE SUM OF THE TWO REMOTE ANGLES. The remote angles are those interior angles that are not adjacent to the exterior angle so in this case ∠1 & ∠2 are the remote angles. B
m∠1 + m∠2 = m∠4, Explain why this would be true. 1
2
A
Answer: m∠3 + m∠4 = 180° (Linear Pair), m∠1 + m∠2 + m∠3 = 180° (Sum of Interior ∠’s of a Δ) m∠1 + m∠2 + m∠3 = m∠3 + m∠4 (substitution), m∠1 + m∠2 = m∠4 (+/-­‐ Property) 4
3
C
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