Question 1(Multiple Choice Worth 1 points)

Question

Question 1(Multiple Choice Worth 1 points)

(02.04 MC)

In ΔABC shown below, Line segment AB is congruent to Line segment BC:

Triangle ABC, where sides AB and CB are congruent

Given: line segment AB≅line segment BC

Prove: The base angles of an isosceles triangle are congruent.

The two-column proof with missing statement proves the base angles of an isosceles triangle are congruent:

Statement Reason
1. segment BD is an angle bisector of ∠ABC. 1. by Construction
2. ∠ABD ≅ ∠CBD 2. Definition of an Angle Bisector
3. 3. Reflexive Property
4. ΔABD ≅ ΔCBD 4. Side-Angle-Side (SAS) Postulate
5. ∠BAC ≅ ∠BCA 5. CPCTC

Which statement can be used to fill in the numbered blank space?
line segment BD ≅ line segment AC
line segment BD ≅ line segment BD
line segment AC ≅ line segment AC
line segment AD ≅ line segment DC
Question 2(Multiple Choice Worth 1 points)
(02.04 MC)

Isosceles triangle ABC contains angle bisectors segment BF, segment AD, and segment CE that intersect at X.

triangle ABC with diagonals BF, AD, and EC that intersect at point X

If segment BA is congruent to segment BC and m∠BCA = 44°, what is m∠CXA?

136°
132°
68°
44°
Question 3(Multiple Choice Worth 1 points)
(02.04 MC)

Find the measure of angle x in the figure below:

Two triangles are shown such that one triangle is inverted and they share a common vertex. The lower triangle has two angles at the base. The lower left angle is marked as 60 degrees. The lower right angle is marked as 50 degrees. The angle at the vertex of the inverted triangle at the top is marked as x degrees. The angle at the vertex of the bottom triangle is marked as y degrees.

60°
50°
110°
70°
Question 4(Multiple Choice Worth 1 points)
(02.04 MC)

Triangle ABC is shown below:

Triangle ABC. Line passes through points D, B, and E.

Given: ΔABC

Prove: All three angles of ΔABC add up to 180°.

The flowchart with missing reason proves the measures of the interior angles of ΔABC total 180°:

Top path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle EBC is congruent to angle BCA. By Substitution, the sum of the measures of angles BCA, CBA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle DBA is congruent to angle BAC. By Substitution, the sum of the measures of angles BCA, BCA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Definition of a Straight Angle, the measure of angle EBD equals 180 degrees. By space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees. Bottom path, by Construction, line segment DE is parallel to line segment AC. By Angle Addition Postulate, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees.

Which reason can be used to fill in the numbered blank space?

Definition of Supplementary Angles
Substitution
Definition of Complementary Angles
Angle Addition Postulate
Question 5(Multiple Choice Worth 1 points)
(02.04 MC)

Find the measure of angle x in the figure below:

A triangle is shown. At the top vertex of the triangle is a horizontal line aligned to the base of the triangle. The angle formed between the horizontal line and the left edge of the triangle is shown as 56 degrees, and the angle formed between the horizontal line and the right edge of the triangle is shown as 51 degrees. The angle at the top vertex of the triangle is labeled as y, and the interior angle on the right is labeled as 72 degrees. The interior angle on the left is labeled as x.

35°
47°
73°
78°
Question 6(Multiple Choice Worth 1 points)
(02.04 LC)

A student states that a triangle can be formed with side lengths 4 in, 5 in, and 8 in. Is the student correct? Why, or why not?

Yes, because 4 + 5 > 8
Yes, because 5 + 8 < 4
No, because 4 + 5 > 8
No, because 5 + 8 > 4
Question 7(Multiple Choice Worth 1 points)
(02.04 MC)

Find the measure of angle x in the figure below:

Two triangles are shown such that one triangle is inverted and share a common vertex. The lower triangle has two angles at the base marked as 55 degrees each. The angle at the vertex of the inverted triangle at the top is marked as x degrees.

65°
70°
110°
125°
Question 8(Multiple Choice Worth 1 points)
(02.04 MC)

Given ΔMNO, find the measure of ∠MNO.

Triangle MNO with segment LM forming a straight angle with segment MO and segment OP forming a straight angle with segment MO, the measure of angle NOP is 104 degrees, and segment MN and NO are marked congruent.

28°
38°
52°
76°
Question 9(Multiple Choice Worth 1 points)
(02.04 LC)

Which of the following possibilities will form a triangle?

Side = 16 cm, side = 8 cm, side = 7 cm
Side = 16 cm, side = 9 cm, side = 7 cm
Side = 17 cm, side = 12 cm, side = 7 cm
Side = 17 cm, side = 11 cm, side = 5 cm
Question 10(Multiple Choice Worth 1 points)
(02.04 MC)

triangle EFG in which segment EF measures 3 units and segment FG measures 5 units

In ΔEFG, is it possible for segment GE to measure 9 units?

Yes, because 3 + 5 > 9, 5 + 9 > 3, and 9 + 3 > 5
No, because 3 + 5 > 9, 5 + 9 > 3, and 9 + 3 > 5
Yes, because 3 + 5 < 9, 5 + 9 > 3, and 9 + 3 > 5
No, because 3 + 5 < 9, 5 + 9 > 3, and 9 + 3 > 5

1 answer

Here are the answers to the questions:

Question 1:
Statement: line segment BD ≅ line segment BD
Reason: Reflexive Property

Question 2:
To find m∠CXA, we first note that triangle ABC is isosceles with AB ≅ BC. Therefore, angles BCA and BAC are also congruent. We can find m∠CBA as follows:
m∠BCA = 44° implies m∠BAC = 44° as well.
Now, in triangle ABC:
m∠ABC = 180° - m∠BAC - m∠BCA = 180° - 44° - 44° = 92°.
Since ∠CXA is the exterior angle for triangle BCA, we have:
m∠CXA = m∠CBA + m∠CAB = 92° + 44° = 136°.
So, the answer is 136°.

Question 3:
In the lower triangle, the sum of the angles is 180°. Thus, if x is the angle at the inverted triangle, we can calculate it as follows:
Sum of angles = 180°
60° + 50° + y = 180°
y = 180° - 110° = 70°.
Therefore, angle x must be the same as angle y, leading to: x = 70°.

Question 4:
The reason that can be used to fill in the numbered blank space is:
Angle Addition Postulate.

Question 5:
The sum of angles in a triangle is 180°. Therefore, we can set up the equation:
56° + 51° + x = 180°
x = 180° - 107° = 73°.
So, the measure of angle x is 73°.

Question 6:
The student is correct:
Yes, because 4 + 5 > 8.

Question 7:
Again, using the triangle angle sum theorem:
55° + 55° + x = 180°
x = 180° - 110° = 70°.
So, the answer is 70°.

Question 8:
To find the measure of ∠MNO, we note that: ∠NOP = 104° and it forms a straight line with ∠MNO. Thus: m∠MNO + m∠NOP = 180°
m∠MNO + 104° = 180°
m∠MNO = 180° - 104° = 76°.
So, the answer is 76°.

Question 9:
To determine which side lengths can form a triangle, we use the triangle inequality theorem. The only valid combination is:
Side = 16 cm, side = 8 cm, side = 7 cm (Since 16 < 8 + 7).

Question 10:
To determine if GE can measure 9 units, we use the triangle inequality.
3 + 5 > 9 is false. Therefore, the correct answer is:
No, because 3 + 5 < 9, 5 + 9 > 3, and 9 + 3 > 5.

These are the answers to the multiple-choice questions, in order.

Ask a New Question or answer this question.

GPT-4o mini · Answer

Here are the answers to the questions: **Question 1:** **Statement:** line segment BD ≅ line segment BD **Reason:** Reflexive Property --- **Question 2:** To find m∠CXA, we first note that triangle ABC is isosceles with AB ≅ BC. Therefore, angles BCA and BAC are also congruent. We can find m∠CBA as follows: m∠BCA = 44° implies m∠BAC = 44° as well. Now, in triangle ABC: m∠ABC = 180° - m∠BAC - m∠BCA = 180° - 44° - 44° = 92°. Since ∠CXA is the exterior angle for triangle BCA, we have: m∠CXA = m∠CBA + m∠CAB = 92° + 44° = 136°. So, the answer is **136°**. --- **Question 3:** In the lower triangle, the sum of the angles is 180°. Thus, if x is the angle at the inverted triangle, we can calculate it as follows: Sum of angles = 180° 60° + 50° + y = 180° y = 180° - 110° = 70°. Therefore, angle x must be the same as angle y, leading to: **x = 70°**. --- **Question 4:** The reason that can be used to fill in the numbered blank space is: **Angle Addition Postulate**. --- **Question 5:** The sum of angles in a triangle is 180°. Therefore, we can set up the equation: 56° + 51° + x = 180° x = 180° - 107° = 73°. So, the measure of angle x is **73°**. --- **Question 6:** The student is correct: **Yes, because 4 + 5 > 8**. --- **Question 7:** Again, using the triangle angle sum theorem: 55° + 55° + x = 180° x = 180° - 110° = 70°. So, the answer is **70°**. --- **Question 8:** To find the measure of ∠MNO, we note that: ∠NOP = 104° and it forms a straight line with ∠MNO. Thus: m∠MNO + m∠NOP = 180° m∠MNO + 104° = 180° m∠MNO = 180° - 104° = 76°. So, the answer is **76°**. --- **Question 9:** To determine which side lengths can form a triangle, we use the triangle inequality theorem. The only valid combination is: **Side = 16 cm, side = 8 cm, side = 7 cm** (Since 16 < 8 + 7). --- **Question 10:** To determine if GE can measure 9 units, we use the triangle inequality. 3 + 5 > 9 is false. Therefore, the correct answer is: **No, because 3 + 5 < 9, 5 + 9 > 3, and 9 + 3 > 5**. --- These are the answers to the multiple-choice questions, in order.