Lines - Math
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Figure NOT drawn to scale.
Refer to the above diagram.
and 
are supplementary.
True, false, or inconclusive: It follows that 
.
Figure NOT drawn to scale.
Refer to the above diagram.
True, false, or inconclusive: It follows that
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and 
form a linear pair of angles and are supplementary regardless of whether or not 
.
If 
, 
, and 
, what is the measure, in degrees, of 
?
If 
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The question states that 
. The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:
The sum of angles of a triangle is equal to 180 degrees. The question states that 
; therefore we know the following measure:
Use this information to solve for the missing angle: 
The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:
The measure of 
is 148 degrees.
The question states that 
The sum of angles of a triangle is equal to 180 degrees. The question states that 
Use this information to solve for the missing angle:
The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:
The measure of 
In the following diagram, lines 
and 
are parallel to each other. What is the value for 
?
In the following diagram, lines
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When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.
Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.
When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.
Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.
Two pairs of parallel lines intersect:
If A = 135o, what is 2*|B-C| = ?
Two pairs of parallel lines intersect:
If A = 135o, what is 2*|B-C| = ?
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By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2*|B-C| = 2* |45-135| = 180o
By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2*|B-C| = 2* |45-135| = 180o
Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.
What is the degree measure of angle 
?
Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.
What is the degree measure of angle
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Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, 
and 
which will sum up to 
. Setting up an algebraic equation for this, we get 
. Solving for 
, we get 
. With this, we can get either 
(for the smaller angle) or 
(for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as 
degrees.
Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles,
Figure not drawn to scale.
In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?
Figure not drawn to scale.
In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?
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Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.
Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:
x + (x + 81) + x = 180
Simplify by collecting the x terms.
3x + 81 = 180
Subtract 81 from both sides.
3x = 99
Divide by 3.
x = 33.
This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.
measure of CPB = 33 + 33 = 66.
The answer is 66.
Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.
Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:
x + (x + 81) + x = 180
Simplify by collecting the x terms.
3x + 81 = 180
Subtract 81 from both sides.
3x = 99
Divide by 3.
x = 33.
This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.
measure of CPB = 33 + 33 = 66.
The answer is 66.
One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?
One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?
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Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.
Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180.
We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:
(1/2)y = 2x.
Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.
(1/2)(180-x) = 2x.
Multiply both sides by 2 to get rid of the fraction.
(180 – x) = 4x.
Add x to both sides.
180 = 5x.
Divide both sides by 5.
x = 36.
The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:
x + z = 90.
Now, we can substitute 36 as the value of x and then solve for z.
36 + z = 90.
Subtract 36 from both sides.
z = 54.
The answer is 54.
Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.
Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180.
We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:
(1/2)y = 2x.
Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.
(1/2)(180-x) = 2x.
Multiply both sides by 2 to get rid of the fraction.
(180 – x) = 4x.
Add x to both sides.
180 = 5x.
Divide both sides by 5.
x = 36.
The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:
x + z = 90.
Now, we can substitute 36 as the value of x and then solve for z.
36 + z = 90.
Subtract 36 from both sides.
z = 54.
The answer is 54.
In the diagram, AB || CD. What is the value of a+b?
In the diagram, AB || CD. What is the value of a+b?
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Refer to the following diagram while reading the explanation:
We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.
Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.
Therefore, a + b = 140° + 20° = 160°
Refer to the following diagram while reading the explanation:
We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.
Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.
Therefore, a + b = 140° + 20° = 160°
The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle A. What is the sum, in degrees, of the measures of the supplement and complement of angle A?
The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle A. What is the sum, in degrees, of the measures of the supplement and complement of angle A?
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Let A represent the measure, in degrees, of angle A. By definition, the sum of the measures of A and its complement is 90 degrees. We can write the following equation to determine an expression for the measure of the complement of angle A.
A + measure of complement of A = 90
Subtract A from both sides.
measure of complement of A = 90 – A
Similarly, because the sum of the measures of angle A and its supplement is 180 degrees, we can represent the measure of the supplement of A as 180 – A.
The problem states that the measure of the supplement of A is 40 degrees larger than twice the measure of the complement of A. We can write this as 2(90-A) + 40.
Next, we must set the two expressions 180 – A and 2(90 – A) + 40 equal to one another and solve for A:
180 – A = 2(90 – A) + 40
Distribute the 2:
180 - A = 180 – 2A + 40
Add 2A to both sides:
180 + A = 180 + 40
Subtract 180 from both sides:
A = 40
Therefore the measure of angle A is 40 degrees.
The question asks us to find the sum of the measures of the supplement and complement of A. The measure of the supplement of A is 180 – A = 180 – 40 = 140 degrees. Similarly, the measure of the complement of A is 90 – 40 = 50 degrees.
The sum of these two is 140 + 50 = 190 degrees.
Let A represent the measure, in degrees, of angle A. By definition, the sum of the measures of A and its complement is 90 degrees. We can write the following equation to determine an expression for the measure of the complement of angle A.
A + measure of complement of A = 90
Subtract A from both sides.
measure of complement of A = 90 – A
Similarly, because the sum of the measures of angle A and its supplement is 180 degrees, we can represent the measure of the supplement of A as 180 – A.
The problem states that the measure of the supplement of A is 40 degrees larger than twice the measure of the complement of A. We can write this as 2(90-A) + 40.
Next, we must set the two expressions 180 – A and 2(90 – A) + 40 equal to one another and solve for A:
180 – A = 2(90 – A) + 40
Distribute the 2:
180 - A = 180 – 2A + 40
Add 2A to both sides:
180 + A = 180 + 40
Subtract 180 from both sides:
A = 40
Therefore the measure of angle A is 40 degrees.
The question asks us to find the sum of the measures of the supplement and complement of A. The measure of the supplement of A is 180 – A = 180 – 40 = 140 degrees. Similarly, the measure of the complement of A is 90 – 40 = 50 degrees.
The sum of these two is 140 + 50 = 190 degrees.
In rectangle ABCD, both diagonals are drawn and intersect at point E.
Let the measure of angle AEB equal x degrees.
Let the measure of angle BEC equal y degrees.
Let the measure of angle CED equal z degrees.
Find the measure of angle AED in terms of x, y, and/or z.
In rectangle ABCD, both diagonals are drawn and intersect at point E.
Let the measure of angle AEB equal x degrees.
Let the measure of angle BEC equal y degrees.
Let the measure of angle CED equal z degrees.
Find the measure of angle AED in terms of x, y, and/or z.
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Intersecting lines create two pairs of vertical angles which are congruent. Therefore, we can deduce that y = measure of angle AED.
Furthermore, intersecting lines create adjacent angles that are supplementary (sum to 180 degrees). Therefore, we can deduce that x + y + z + (measure of angle AED) = 360.
Substituting the first equation into the second equation, we get
x + (measure of angle AED) + z + (measure of angle AED) = 360
2(measure of angle AED) + x + z = 360
2(measure of angle AED) = 360 – (x + z)
Divide by two and get:
measure of angle AED = 180 – 1/2(x + z)
Intersecting lines create two pairs of vertical angles which are congruent. Therefore, we can deduce that y = measure of angle AED.
Furthermore, intersecting lines create adjacent angles that are supplementary (sum to 180 degrees). Therefore, we can deduce that x + y + z + (measure of angle AED) = 360.
Substituting the first equation into the second equation, we get
x + (measure of angle AED) + z + (measure of angle AED) = 360
2(measure of angle AED) + x + z = 360
2(measure of angle AED) = 360 – (x + z)
Divide by two and get:
measure of angle AED = 180 – 1/2(x + z)
AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is 
, what is the measure of angle 2?
AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is
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The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.
The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.
$\overline{AB}$ is a straight line. $\overline{CD}$ intersects $\overline{AB}$ at point E. If angle AEC measures 120 degrees, what must be the measure of angle BEC?
$\overline{AB}$ is a straight line. $\overline{CD}$ intersects $\overline{AB}$ at point E. If angle AEC measures 120 degrees, what must be the measure of angle BEC?
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angle AEC & angle BEC must add up to 180 degrees. So, if angle AEC is 120, angle BEC (the supplementary angle) must equal 60, for a total of 180.
angle AEC & angle BEC must add up to 180 degrees. So, if angle AEC is 120, angle BEC (the supplementary angle) must equal 60, for a total of 180.
If angle A measures $(40-10x)^{circ}$, which of the following is equivalent to the measure of the supplement of angle A ?
If angle A measures $(40-10x)^{circ}$, which of the following is equivalent to the measure of the supplement of angle A ?
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When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:
measure of angle A+ measure of supplement of angle A=180
40-10x+ measure of supplement of angle A=180
Subtract 40 from both sides.
-10x+ measure of supplement of angle A=140
Add 10x to both sides.
measure of supplement of angle A=140+10x=10x+140
The answer is $(10x+140)^{circ}$.
When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:
measure of angle A+ measure of supplement of angle A=180
40-10x+ measure of supplement of angle A=180
Subtract 40 from both sides.
-10x+ measure of supplement of angle A=140
Add 10x to both sides.
measure of supplement of angle A=140+10x=10x+140
The answer is $(10x+140)^{circ}$.
Lines 
and 
are parallel. 
, 
, 
is a right triangle, and 
has a length of 10. What is the length of 
Lines
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Since we know opposite angles are equal, it follows that angle 
and 
.
Imagine a parallel line passing through point 
. The imaginary line would make opposite angles with 
& 
, the sum of which would equal 
. Therefore, 
.
cos (60)=.5=\frac{EG}{EF}$rightarrow EF=\frac{EG}{.5}$=20
Since we know opposite angles are equal, it follows that angle
Imagine a parallel line passing through point
cos (60)=.5=\frac{EG}{EF}$rightarrow EF=\frac{EG}{.5}$=20
Two angles are supplementary and have a ratio of 1:4. What is the size of the smaller angle?
Two angles are supplementary and have a ratio of 1:4. What is the size of the smaller angle?
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Since the angles are supplementary, their sum is 180 degrees. Because they are in a ratio of 1:4, the following expression could be written:
x+4x=180
5x=180
$x=36^{circ}$
Since the angles are supplementary, their sum is 180 degrees. Because they are in a ratio of 1:4, the following expression could be written:
x+4x=180
5x=180
$x=36^{circ}$
A student creates a challenge for his friend. He first draws a square, the adds the line for each of the 2 diagonals. Finally, he asks his friend to draw the circle that has the most intersections possible.
How many intersections will this circle have?
A student creates a challenge for his friend. He first draws a square, the adds the line for each of the 2 diagonals. Finally, he asks his friend to draw the circle that has the most intersections possible.
How many intersections will this circle have?
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The answer to this problem is 12. This can be drawn as shown below (intersections marked in red).
We can also be sure that this is the maximal case because it is the largest answer selection. Were it not given as a multiple choice question, however, we could still be sure this was the largest. This is because no line can intersect a circle in more than 2 points. Keeping this in mind, we look at the construction of our initial shape. The square has 4 lines, and then each diagonal is an additional 2. We have thus drawn in 6 lines. The maximum number of intersections is therefore going to be twice this, or 12.
The answer to this problem is 12. This can be drawn as shown below (intersections marked in red).
We can also be sure that this is the maximal case because it is the largest answer selection. Were it not given as a multiple choice question, however, we could still be sure this was the largest. This is because no line can intersect a circle in more than 2 points. Keeping this in mind, we look at the construction of our initial shape. The square has 4 lines, and then each diagonal is an additional 2. We have thus drawn in 6 lines. The maximum number of intersections is therefore going to be twice this, or 12.
Examine the diagram. Which of these conditions does not prove that 
?
Examine the diagram. Which of these conditions does not prove that
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If 
and 
, then 
, since two lines parallel to the same line are parallel to each other.
If 
, then 
, since two same-side interior angles formed by transversal 
are supplementary.
If 
, then 
, since two alternate interior angles formed by transversal 
are congruent.
However, 
regardless of whether 
and 
are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.
If
If
If
However,
An isosceles triangle has an interior angle that measures 
. What are the measures of its other two angles?
An isosceles triangle has an interior angle that measures
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By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is 
, each of the missing angles measures 
.
By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is
What is the measure of 
?
What is the measure of
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When two parallel lines are crossed by a third line (called the transversal), the measure of the angles follows a specific pattern. The pairs of angles inside the two lines and on opposite sides are called alternate interior angles. Alternate interior angles, such as 
and 
, have the same degree measure. Therefore, the measure of 
is 
.
When two parallel lines are crossed by a third line (called the transversal), the measure of the angles follows a specific pattern. The pairs of angles inside the two lines and on opposite sides are called alternate interior angles. Alternate interior angles, such as
How would you classify the following angle?
How would you classify the following angle?
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Obtuse angles are greater than 
.
Scalene is a designation for triangles that have one angle greater than 
, but this figure is not a triangle.
Acute angles are less than 
, right angles are 
, and straight angles are 
.
Therefore this angle is obtuse.
Obtuse angles are greater than
Scalene is a designation for triangles that have one angle greater than
Acute angles are less than
Therefore this angle is obtuse.