sas what do we already know and what do we need the know
36 answers
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Bill knows that one angle in an isosceles triangle measures 100 degrees. If the other 2 angles are congruent, what is the measure of each of those angles?
In an isosceles triangle, the two congruent angles are equal to each other. Let's call that angle "x". According to the problem, we know that one angle in the triangle, which is not congruent to the other two, measures 100 degrees. We can use this information to set up an equation:
100 + x + x = 180
Simplifying and solving for x, we get:
100 + 2x = 180
2x = 80
x = 40
Therefore, each of the congruent angles measures 40 degrees in the isosceles triangle.
100 + x + x = 180
Simplifying and solving for x, we get:
100 + 2x = 180
2x = 80
x = 40
Therefore, each of the congruent angles measures 40 degrees in the isosceles triangle.
Elise is measuring the angles of a triangle. She measures the first two and gets 30° and 100°. What should be the measure of the third angle?
The sum of the angles in a triangle is always 180 degrees. If we add the given angles 30° and 100° together, we get:
30° + 100° = 130°
To find the measure of the third angle, we can subtract 130° from 180°:
180° - 130° = 50°
Therefore, the measure of the third angle should be 50 degrees.
30° + 100° = 130°
To find the measure of the third angle, we can subtract 130° from 180°:
180° - 130° = 50°
Therefore, the measure of the third angle should be 50 degrees.
In ΔMNO m∠O < m∠M < m∠N. List the side lengths in order from shortest to longest.
We cannot determine the order of the side lengths in the triangle based on the given information.
However, we can infer that the side opposite the smallest angle (angle O) will be the shortest side, and the side opposite the largest angle (angle N) will be the longest side.
Without additional information about the size of each angle, we cannot determine the order of the side lengths beyond that.
However, we can infer that the side opposite the smallest angle (angle O) will be the shortest side, and the side opposite the largest angle (angle N) will be the longest side.
Without additional information about the size of each angle, we cannot determine the order of the side lengths beyond that.
In ΔMNO m∠O < m∠M < m∠N. List the side lengths in order from shortest to longest.
Is MN, NO, MO correct??
Is MN, MO, NO Correct
is NO, MN, MO correct
Is MO, NO, MN Correct
Is MN, NO, MO correct??
Is MN, MO, NO Correct
is NO, MN, MO correct
Is MO, NO, MN Correct
The correct answer is:
MO, NO, MN.
The side opposite the smallest angle (angle O) is MO, which is the shortest side. The side opposite the largest angle (angle N) is MN, which is the longest side. And NO is the side opposite the remaining angle (angle M).
MO, NO, MN.
The side opposite the smallest angle (angle O) is MO, which is the shortest side. The side opposite the largest angle (angle N) is MN, which is the longest side. And NO is the side opposite the remaining angle (angle M).
By the Triangle Inequality Theorem which set of side lengths could create a triangle?
10, 4, 5
5, 7, 13
7, 9, 3
5, 10, 4
10, 4, 5
5, 7, 13
7, 9, 3
5, 10, 4
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, a set of side lengths can create a triangle if and only if it satisfies this condition.
Applying this rule to the options given, we find that only the set of side lengths 5, 10, 4 satisfies the Triangle Inequality Theorem:
- 10 < 4 + 5 is false
- 5 + 7 > 13 is true
- 7 + 9 > 3 is false
- 5 + 4 > 10 is true
Therefore, the only set of side lengths that could create a triangle is 5, 10, 4.
Applying this rule to the options given, we find that only the set of side lengths 5, 10, 4 satisfies the Triangle Inequality Theorem:
- 10 < 4 + 5 is false
- 5 + 7 > 13 is true
- 7 + 9 > 3 is false
- 5 + 4 > 10 is true
Therefore, the only set of side lengths that could create a triangle is 5, 10, 4.
could these be the measures of 3 angles of a triangle?
60 degrees, 60 degrees, 60 degrees 60 degrees, 60 degrees, 60 degrees 100 degrees, 40 degrees, 30 degrees 100 degrees, 40 degrees, 30 degrees 50 degrees, 60 degrees, 50 degrees 50 degrees, 60 degrees, 50 degrees 10 degrees, 60 degrees, 100 degrees
60 degrees, 60 degrees, 60 degrees 60 degrees, 60 degrees, 60 degrees 100 degrees, 40 degrees, 30 degrees 100 degrees, 40 degrees, 30 degrees 50 degrees, 60 degrees, 50 degrees 50 degrees, 60 degrees, 50 degrees 10 degrees, 60 degrees, 100 degrees
According to the Triangle Sum Theorem, the sum of the measures of the angles in a triangle is 180 degrees. Therefore, we can apply this rule to check if each set of angles could be the measures of the angles of a triangle:
- 60 degrees, 60 degrees, 60 degrees: This is a valid set of angles and is the measure of the angles of an equilateral triangle.
- 60 degrees, 60 degrees, 60 degrees: This is also a valid set of angles and is the measure of the angles of an equilateral triangle.
- 100 degrees, 40 degrees, 30 degrees: This is a valid set of angles and could be the measure of the angles of a triangle.
- 100 degrees, 40 degrees, 30 degrees: This is also a valid set of angles and could be the measure of the angles of a triangle.
- 50 degrees, 60 degrees, 50 degrees: This is a valid set of angles and could be the measure of the angles of a triangle.
- 50 degrees, 60 degrees, 50 degrees: This is also a valid set of angles and could be the measure of the angles of a triangle.
- 10 degrees, 60 degrees, 100 degrees: This is an invalid set of angles because its sum is less than 180 degrees.
Therefore, the sets of angles that could be the measures of angles of a triangle are:
- 60 degrees, 60 degrees, 60 degrees
- 60 degrees, 60 degrees, 60 degrees
- 100 degrees, 40 degrees, 30 degrees
- 100 degrees, 40 degrees, 30 degrees
- 50 degrees, 60 degrees, 50 degrees
- 50 degrees, 60 degrees, 50 degrees
- 60 degrees, 60 degrees, 60 degrees: This is a valid set of angles and is the measure of the angles of an equilateral triangle.
- 60 degrees, 60 degrees, 60 degrees: This is also a valid set of angles and is the measure of the angles of an equilateral triangle.
- 100 degrees, 40 degrees, 30 degrees: This is a valid set of angles and could be the measure of the angles of a triangle.
- 100 degrees, 40 degrees, 30 degrees: This is also a valid set of angles and could be the measure of the angles of a triangle.
- 50 degrees, 60 degrees, 50 degrees: This is a valid set of angles and could be the measure of the angles of a triangle.
- 50 degrees, 60 degrees, 50 degrees: This is also a valid set of angles and could be the measure of the angles of a triangle.
- 10 degrees, 60 degrees, 100 degrees: This is an invalid set of angles because its sum is less than 180 degrees.
Therefore, the sets of angles that could be the measures of angles of a triangle are:
- 60 degrees, 60 degrees, 60 degrees
- 60 degrees, 60 degrees, 60 degrees
- 100 degrees, 40 degrees, 30 degrees
- 100 degrees, 40 degrees, 30 degrees
- 50 degrees, 60 degrees, 50 degrees
- 50 degrees, 60 degrees, 50 degrees
Which side lengths of a triangle would describe an isosceles triangle?
5 cm, 12 cm, 13 cm5 cm, 12 cm, 13 cm
3 cm, 4 cm, 5 cm3 cm, 4 cm, 5 cm
6 cm, 6 cm, 6 cm6 cm, 6 cm, 6 cm
5 cm, 8 cm, 5 cm
5 cm, 12 cm, 13 cm5 cm, 12 cm, 13 cm
3 cm, 4 cm, 5 cm3 cm, 4 cm, 5 cm
6 cm, 6 cm, 6 cm6 cm, 6 cm, 6 cm
5 cm, 8 cm, 5 cm
The side lengths that would describe an isosceles triangle are:
6 cm, 6 cm, 6 cm (same length for all sides)
OR
5 cm, 8 cm, 5 cm (two sides have the same length, in this case both sides with 5 cm)
The other options do not describe isosceles triangles. The first option is a Pythagorean triple (meaning its sides satisfy the Pythagorean Theorem), while the second option has no side lengths the same, so it cannot be isosceles.
6 cm, 6 cm, 6 cm (same length for all sides)
OR
5 cm, 8 cm, 5 cm (two sides have the same length, in this case both sides with 5 cm)
The other options do not describe isosceles triangles. The first option is a Pythagorean triple (meaning its sides satisfy the Pythagorean Theorem), while the second option has no side lengths the same, so it cannot be isosceles.
Which set of angle measures CANNOT be the angle measures of a triangle?
13.9 degrees, 16.1 degrees, 150 degrees13.9 degrees, 16.1 degrees, 150 degrees
1.5 degrees, 1.5 degrees, 177 degrees1.5 degrees, 1.5 degrees, 177 degrees
50 degrees,49.1 degrees, 80 degrees50 degrees,49.1 degrees, 80 degrees
58 degrees, 61 degrees, 61 degrees
13.9 degrees, 16.1 degrees, 150 degrees13.9 degrees, 16.1 degrees, 150 degrees
1.5 degrees, 1.5 degrees, 177 degrees1.5 degrees, 1.5 degrees, 177 degrees
50 degrees,49.1 degrees, 80 degrees50 degrees,49.1 degrees, 80 degrees
58 degrees, 61 degrees, 61 degrees
According to the Triangle Inequality Theorem, the sum of the measures of any two angles of a triangle must be greater than the measure of the third angle. Therefore, a set of angle measures cannot be the angle measures of a triangle if and only if it violates this condition.
Considering the angle measures given in each option, we can apply this rule to identify which option is not the angle measures of a triangle:
- 13.9 degrees, 16.1 degrees, 150 degrees: This is a valid set of angle measures which satisfies the Triangle Inequality Theorem and could be the angle measures of a triangle. The third angle is just an obtuse angle.
- 1.5 degrees, 1.5 degrees, 177 degrees: This option does not satisfy the Triangle Inequality Theorem, since the first two angles have a sum of only 3 degrees, which is less than the third angle of 177 degrees. Therefore, this option cannot be the angle measures of a triangle.
- 50 degrees, 49.1 degrees, 80 degrees: This is a valid set of angle measures which satisfies the Triangle Inequality Theorem and could be the angle measures of a triangle.
- 58 degrees, 61 degrees, 61 degrees: This is not a valid set of angle measures because it violates the rule that the sum of the measures of any two angles of a triangle must be greater than the measure of the third angle. In this case, the measure of the first angle (58 degrees) plus the measure of the second angle (61 degrees) is only 119 degrees, which is less than the measure of the third angle (61 degrees). Therefore, this set of angle measures cannot be the angle measures of a triangle.
Therefore, the set of angle measures that CANNOT be the angle measures of a triangle is 58 degrees, 61 degrees, 61 degrees.
Considering the angle measures given in each option, we can apply this rule to identify which option is not the angle measures of a triangle:
- 13.9 degrees, 16.1 degrees, 150 degrees: This is a valid set of angle measures which satisfies the Triangle Inequality Theorem and could be the angle measures of a triangle. The third angle is just an obtuse angle.
- 1.5 degrees, 1.5 degrees, 177 degrees: This option does not satisfy the Triangle Inequality Theorem, since the first two angles have a sum of only 3 degrees, which is less than the third angle of 177 degrees. Therefore, this option cannot be the angle measures of a triangle.
- 50 degrees, 49.1 degrees, 80 degrees: This is a valid set of angle measures which satisfies the Triangle Inequality Theorem and could be the angle measures of a triangle.
- 58 degrees, 61 degrees, 61 degrees: This is not a valid set of angle measures because it violates the rule that the sum of the measures of any two angles of a triangle must be greater than the measure of the third angle. In this case, the measure of the first angle (58 degrees) plus the measure of the second angle (61 degrees) is only 119 degrees, which is less than the measure of the third angle (61 degrees). Therefore, this set of angle measures cannot be the angle measures of a triangle.
Therefore, the set of angle measures that CANNOT be the angle measures of a triangle is 58 degrees, 61 degrees, 61 degrees.
What is the measure of the angle ∠QPR, between the ramp and the ground, if angle ∠RQP is a right angle?
Responses
36°36°
23°23°
17°17°
31°31°
27°
Responses
36°36°
23°23°
17°17°
31°31°
27°
If ∠RQP is a right angle, then we know that it measures 90 degrees. Since triangle PQR must have a total of 180 degrees, we can use the fact that the sum of angles in a triangle is 180 degrees to find the measure of ∠QPR:
180 - 90 - 36 = 54
Therefore, ∠QPR measures 54 degrees.
180 - 90 - 36 = 54
Therefore, ∠QPR measures 54 degrees.
The ramp in the picture makes an angle of 73° with the doorpost.What is the measure of the angle ∠QPR, between the ramp and the ground, if angle ∠RQP is a right angle?
Responses
36°
23°
17°
31°
27°
Responses
36°
23°
17°
31°
27°
If the ramp makes an angle of 73° with the doorpost and ∠RQP is a right angle (i.e., 90 degrees), then we can use trigonometry to find the measure of the angle ∠QPR.
We know that tan(73°) = PR/QP, where PR is the length of the ramp and QP is the length of the horizontal distance between the doorpost and the base of the ramp. Therefore, we can solve for PR/QP:
tan(73°) = PR/QP
PR = QP * tan(73°)
Next, we can use the Pythagorean theorem to find the length of QR:
QR^2 = PR^2 + PQ^2
QR^2 = (QP * tan(73°))^2 + PQ^2
QR = sqrt((QP * tan(73°))^2 + PQ^2)
Finally, we can use the definition of sine to find the measure of the angle ∠QPR:
sin(∠QPR) = PQ/QR
∠QPR = arcsin(PQ/QR)
Note that the PQ/QR ratio can also be expressed as sin(90° - 73°) = cos(73°).
Substituting the known values, we get:
QR = sqrt((QP * tan(73°))^2 + PQ^2) = sqrt((x * tan(73°))^2 + y^2)
sin(∠QPR) = PQ/QR = y/sqrt((x * tan(73°))^2 + y^2)
∠QPR = arcsin(y/sqrt((x * tan(73°))^2 + y^2)) = arcsin(sin(90° - 73°)) = arcsin(cos(73°))
Therefore, the measure of the angle ∠QPR is approximately 17 degrees (rounded to the nearest whole number).
Hence, the answer is 17°.
We know that tan(73°) = PR/QP, where PR is the length of the ramp and QP is the length of the horizontal distance between the doorpost and the base of the ramp. Therefore, we can solve for PR/QP:
tan(73°) = PR/QP
PR = QP * tan(73°)
Next, we can use the Pythagorean theorem to find the length of QR:
QR^2 = PR^2 + PQ^2
QR^2 = (QP * tan(73°))^2 + PQ^2
QR = sqrt((QP * tan(73°))^2 + PQ^2)
Finally, we can use the definition of sine to find the measure of the angle ∠QPR:
sin(∠QPR) = PQ/QR
∠QPR = arcsin(PQ/QR)
Note that the PQ/QR ratio can also be expressed as sin(90° - 73°) = cos(73°).
Substituting the known values, we get:
QR = sqrt((QP * tan(73°))^2 + PQ^2) = sqrt((x * tan(73°))^2 + y^2)
sin(∠QPR) = PQ/QR = y/sqrt((x * tan(73°))^2 + y^2)
∠QPR = arcsin(y/sqrt((x * tan(73°))^2 + y^2)) = arcsin(sin(90° - 73°)) = arcsin(cos(73°))
Therefore, the measure of the angle ∠QPR is approximately 17 degrees (rounded to the nearest whole number).
Hence, the answer is 17°.
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°
B 130°
C 55°
D 85°
E 40°
Responses
A 65°
B 130°
C 55°
D 85°
E 40°
In an isosceles triangle, as two sides are equal, the angles opposite the equal sides must also be equal. Therefore, if angle A and angle C are equal in isosceles triangle ABC, then we can set up an equation:
A + B + C = 180
Substituting A = C, we get:
A + B + A = 180
Simplifying, we get:
2A + B = 180
Since the angle measures in a triangle sum up to 180 degrees, we know that A + B + C = 180.
Since triangle ABC is isosceles, A = C and we get:
A + B + A = 180
2A + B = 180
Now, we may be given the value of A or C to directly find the value of B. In the absence of such information, we cannot determine the value of B. Therefore, only knowing that triangle ABC is isosceles is not enough to find the measure of angle B.
A + B + C = 180
Substituting A = C, we get:
A + B + A = 180
Simplifying, we get:
2A + B = 180
Since the angle measures in a triangle sum up to 180 degrees, we know that A + B + C = 180.
Since triangle ABC is isosceles, A = C and we get:
A + B + A = 180
2A + B = 180
Now, we may be given the value of A or C to directly find the value of B. In the absence of such information, we cannot determine the value of B. Therefore, only knowing that triangle ABC is isosceles is not enough to find the measure of angle B.
Jeremy was trying to find the sum of the measures of the angles in a triangle. He cut off all three angles on the dotted lines shown and then laid the angles vertices edge-to-edge. He observed that they formed a straight line. What does this mean?
Responses
A That the angles sum up to 90°.That the angles sum up to 90°.
B Nothing. A straight line has no measure.Nothing. A straight line has no measure.
C That the angles sum up to 270°.That the angles sum up to 270°.
D That the angles sum up to 180°.
Responses
A That the angles sum up to 90°.That the angles sum up to 90°.
B Nothing. A straight line has no measure.Nothing. A straight line has no measure.
C That the angles sum up to 270°.That the angles sum up to 270°.
D That the angles sum up to 180°.
The fact that the three angles fit edge-to-edge to form a straight line means that their measures add up to 180 degrees. This is because a straight line forms an angle measuring 180 degrees. Therefore, the correct answer is D: That the angles sum up to 180°.
Bill knows that one angle in an isosceles triangle measures 100 degrees. If the other 2 angles are congruent, what is the measure of each of those angles?
Responses
A 100 degrees100 degrees
B 40 degrees40 degrees
C 80 degrees80 degrees
D 60 degrees
Responses
A 100 degrees100 degrees
B 40 degrees40 degrees
C 80 degrees80 degrees
D 60 degrees
Let the measure of each of the congruent angles be "x". According to the problem, we know that one angle in the triangle is 100 degrees, and the other two angles are congruent. We can use the fact that the sum of the measures of the angles in a triangle is 180 degrees to set up an equation:
100 + x + x = 180
Simplifying and solving for x, we get:
100 + 2x = 180
2x = 80
x = 40
Therefore, each of the congruent angles in the isosceles triangle has a measure of 40 degrees.
Hence, the correct answer is B: 40 degrees.
100 + x + x = 180
Simplifying and solving for x, we get:
100 + 2x = 180
2x = 80
x = 40
Therefore, each of the congruent angles in the isosceles triangle has a measure of 40 degrees.
Hence, the correct answer is B: 40 degrees.
Elise is measuring the angles of a triangle. She measures the first two and gets 30° and 100°. What should be the measure of the third angle?
Responses
A 130°130°
B 60°60°
C 230°230°
D 50°
Responses
A 130°130°
B 60°60°
C 230°230°
D 50°
According to the triangle sum theorem, the sum of the measures of the angles in a triangle is always 180 degrees. Therefore, to find the measure of the third angle, we need to subtract the measures of the first two angles from 180.
180 - 30 - 100 = 50
Therefore, the measure of the third angle is 50 degrees.
Hence, the correct answer is D: 50°.
180 - 30 - 100 = 50
Therefore, the measure of the third angle is 50 degrees.
Hence, the correct answer is D: 50°.
A carpenter designs an A-frame chalkboard. Its support bars form the legs of an isosceles triangle.
If the measure of base angle ABC is 75°, what is m∠BAC?
Responses
A 210°210°
B 105°105°
C 15°15°
D 30°30°
E 75°75°
If the measure of base angle ABC is 75°, what is m∠BAC?
Responses
A 210°210°
B 105°105°
C 15°15°
D 30°30°
E 75°75°
Since angles BAC and BCA are equal in an isosceles triangle, let's call both angles x.
According to the problem, the measure of angle ABC is 75 degrees. Therefore, by the triangle sum theorem, the sum of angles BAC, BCA, and ABC must be 180 degrees. We can use this information to set up an equation:
x + x + 75 = 180
Simplifying and solving for x, we get:
2x + 75 = 180
2x = 105
x = 52.5
Therefore, the measure of angle BAC is 52.5 degrees.
Hence, the correct answer is not among the options provided.
According to the problem, the measure of angle ABC is 75 degrees. Therefore, by the triangle sum theorem, the sum of angles BAC, BCA, and ABC must be 180 degrees. We can use this information to set up an equation:
x + x + 75 = 180
Simplifying and solving for x, we get:
2x + 75 = 180
2x = 105
x = 52.5
Therefore, the measure of angle BAC is 52.5 degrees.
Hence, the correct answer is not among the options provided.
j = 5 cm , k = 7 cm, and l = 10 cm
In the triangle shown, which shows the correct order of the angle measures from largest to smallest?
Note: Image is not drawn to scale.
Responses
A C, B, A,
B A, C, B
C B, A, C
D C, A, B
In the triangle shown, which shows the correct order of the angle measures from largest to smallest?
Note: Image is not drawn to scale.
Responses
A C, B, A,
B A, C, B
C B, A, C
D C, A, B
To determine the order of the angles from largest to smallest, we need to use the law of cosines:
cos(A) = [b^2 + c^2 - a^2] / 2bc
cos(B) = [a^2 + c^2 - b^2] / 2ac
cos(C) = [a^2 + b^2 - c^2] / 2ab
Substituting the given values, we get:
cos(A) = [7^2 + 10^2 - 5^2] / (2 * 7 * 10) ≈ 0.671
cos(B) = [5^2 + 10^2 - 7^2] / (2 * 5 * 10) = 0.45
cos(C) = [5^2 + 7^2 - 10^2] / (2 * 5 * 7) = 0.05
We can use a calculator to find the inverse cosine of each value to get the measure of each angle:
A ≈ 47.485 degrees
B ≈ 62.931 degrees
C ≈ 69.584 degrees
Therefore, the correct order of the angle measures from largest to smallest is:
C, B, A
Hence, the correct answer
cos(A) = [b^2 + c^2 - a^2] / 2bc
cos(B) = [a^2 + c^2 - b^2] / 2ac
cos(C) = [a^2 + b^2 - c^2] / 2ab
Substituting the given values, we get:
cos(A) = [7^2 + 10^2 - 5^2] / (2 * 7 * 10) ≈ 0.671
cos(B) = [5^2 + 10^2 - 7^2] / (2 * 5 * 10) = 0.45
cos(C) = [5^2 + 7^2 - 10^2] / (2 * 5 * 7) = 0.05
We can use a calculator to find the inverse cosine of each value to get the measure of each angle:
A ≈ 47.485 degrees
B ≈ 62.931 degrees
C ≈ 69.584 degrees
Therefore, the correct order of the angle measures from largest to smallest is:
C, B, A
Hence, the correct answer
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Question
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°
B 130°
C 55°
D 85°
E 40°
29:52
Skip to resources
Question
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°
B 130°
C 55°
D 85°
E 40°
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°65°
B 130°130°
C 55°55°
D 85°85°
E 40°
Responses
A 65°65°
B 130°130°
C 55°55°
D 85°85°
E 40°
Bot answer this question
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°
B 130°
C 55°
D 85°
E 40°
Find the measure of angle B in isosceles triangle ABC.
Responses
A 65°
B 130°
C 55°
D 85°
E 40°