Use Special Triangles To Make Deductions - Trigonometry
Card 1 of 40
Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
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Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is 
. Therefore, 
.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is 
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is 
. Therefore, 
.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is 
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is 
. Therefore, 
.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is 
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In the figure below, 
is inscribed in a circle. 
passes through the center of the circle. In 
, the measure of 
is twice the measure of 
. The figure is drawn to scale.
Which of the following is true about the figure?
In the figure below, 
Which of the following is true about the figure?
Tap to reveal answer
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because 
is a diameter of the circle, arc 
has a measure of 180 degrees. Therefore, 
must be equal to 
. Since 
is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of 
is twice the measure of 
, 
. Therefore, the measure of 
can be calculated as follows:
Therefore, 
is equal to 
. 
must be a 30-60-90 triangle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because 
Therefore, 
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In the figure below, 
is inscribed in a circle. 
passes through the center of the circle. In 
, the measure of 
is twice the measure of 
. The figure is drawn to scale.
Which of the following is true about the figure?
In the figure below, 
Which of the following is true about the figure?
Tap to reveal answer
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because 
is a diameter of the circle, arc 
has a measure of 180 degrees. Therefore, 
must be equal to 
. Since 
is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of 
is twice the measure of 
, 
. Therefore, the measure of 
can be calculated as follows:
Therefore, 
is equal to 
. 
must be a 30-60-90 triangle. Therefore, side length 
must be half the length of side length 
, the hypotenuse of the triangle. Since 
is a diameter of the circle, half of 
represents the length of a radius of the circle. Therefore, 
is equal in length to a radius of the circle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because 
Therefore, 
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In the figure below, 
is a diagonal of quadrilateral 
. 
has a length of 1. 
and 
are congruent and isosceles. 
and 
are perpendicular. The figure is drawn to scale.
Which of the following is a true statement?
In the figure below, 
Which of the following is a true statement?
Tap to reveal answer
Since 
and 
are perpendicular, 
is a right angle. Since no triangle can have more than one right angle, and 
is isosceles, 
must be congruent to 
. Since 
is congruent to 
and 
measures 90 degrees, 
and 
can be calculated as follows:
Therefore, 
and 
are both equal to 45 degrees. 
is a 45-45-90 triangle. Since 
is congruent to 
, 
is also a 45-45-90 triangle. The figure is drawn to scale, so 
is a right angle. Since 
has the same angle measure as 
, the two angles are alternate interior angles and diagonal 
is a transversal relative to 
and 
, which must be parallel.
Since 
Therefore, 
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In the figure below, 
is a diagonal of quadrilateral 
. 
has a length of 
. 
is congruent to 
.
Which of the following is a true statement?
In the figure below, 
Which of the following is a true statement?
Tap to reveal answer
Since 
and 
are perpendicular, 
is a right angle. Since no triangle can have more than one right angle, and 
is isosceles, 
must be congruent to 
. Since angle CBD is congruent to 
and 
measures 90 degrees, 
and 
can be calculated as follows:
Therefore, 
and 
are both equal to 45 degrees. 
is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse 
is 
. Anyone of the four side lengths of quadrilateral 
must, therefore, be equal to 
. To find the area of 
, multiply two side lengths: 
.
Since 
Therefore, 
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between the two short side lengths is 1:1. Therefore, A = B. Triangles with two congruent side lengths are isosceles by definition.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is . Therefore, .
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the shortest side length and the longer non-hypotenuse side length is
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is . Therefore, .
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse length and the second-longest side length is
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the two shorter side lengths are equal. Therefore, A = B.
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Which of the following is true about the right triangle below?
Which of the following is true about the right triangle below?
Tap to reveal answer
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is . Therefore, .
Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 45 - 90 = 45. The pictured triangle is therefore a 45-45-90 triangle. In a 45-45-90 triangle, the ratio between a short side length and the hypotenuse is
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In the figure below, is inscribed in a circle. passes through the center of the circle. In , the measure of is twice the measure of . The figure is drawn to scale.
Which of the following is true about the figure?
In the figure below,
Which of the following is true about the figure?
Tap to reveal answer
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc has a measure of 180 degrees. Therefore, must be equal to . Since is a right triangle, the sum of its interior angles equal 180 degrees. Since the measure of is twice the measure of , . Therefore, the measure of can be calculated as follows:
Therefore, is equal to . must be a 30-60-90 triangle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because
Therefore,
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In the figure below, is inscribed in a circle. passes through the center of the circle. In , the measure of is twice the measure of . The figure is drawn to scale.
Which of the following is true about the figure?
In the figure below,
Which of the following is true about the figure?
Tap to reveal answer
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because is a diameter of the circle, arc has a measure of 180 degrees. Therefore, must be equal to . Since is a right triangle, the sum of its interior angles to 180 degrees. Since the measure of is twice the measure of , . Therefore, the measure of can be calculated as follows:
Therefore, is equal to . must be a 30-60-90 triangle. Therefore, side length must be half the length of side length , the hypotenuse of the triangle. Since is a diameter of the circle, half of represents the length of a radius of the circle. Therefore, is equal in length to a radius of the circle.
For any angle inscribed in a circle, the measure of the angle is equal to half of the resulting arc measure. Because
Therefore,
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In the figure below, is a diagonal of quadrilateral . has a length of 1. and are congruent and isosceles. and are perpendicular. The figure is drawn to scale.
Which of the following is a true statement?
In the figure below,
Which of the following is a true statement?
Tap to reveal answer
Since and are perpendicular, is a right angle. Since no triangle can have more than one right angle, and is isosceles, must be congruent to . Since is congruent to and measures 90 degrees, and can be calculated as follows:
Therefore, and are both equal to 45 degrees. is a 45-45-90 triangle. Since is congruent to , is also a 45-45-90 triangle. The figure is drawn to scale, so is a right angle. Since has the same angle measure as , the two angles are alternate interior angles and diagonal is a transversal relative to and , which must be parallel.
Since
Therefore,
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In the figure below, is a diagonal of quadrilateral . has a length of . is congruent to .
Which of the following is a true statement?
In the figure below,
Which of the following is a true statement?
Tap to reveal answer
Since and are perpendicular, is a right angle. Since no triangle can have more than one right angle, and is isosceles, must be congruent to . Since angle CBD is congruent to and measures 90 degrees, and can be calculated as follows:
Therefore, and are both equal to 45 degrees. is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse is . Anyone of the four side lengths of quadrilateral must, therefore, be equal to . To find the area of , multiply two side lengths: .
Since
Therefore,
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