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In this lesson, we will study relationships between two different angles: that is, geometric reasons why you can determine the measure of one angle from the measure of another angle.
When we looked at angles in triangles, there was only one angle at each vertex, so we could name the angle by giving the name of the vertex (for example, $\cl"red"{∠A}$). The figure to the left shows two different angles with the same vertex $\cl"red"A$: one going from $\cl"red"B$ to $\cl"red"A$ to $\cl"red"C$, and the other going from $\cl"red"C$ to $\cl"red"A$ to $\cl"red"D$. In order to tell them apart, we’ll call the first one of these $\cl"red"{∠BAC}$ and the second one $\cl"red"{∠CAD}$.
Click to add a slider below the figure to the left. This slider changes the sizes of $\cl"red"{∠BAC}$ and $\cl"red"{∠CAD}$, but keeps $\cl"red"B$, $\cl"red"A$, and $\cl"red"D$ all on the same straight line.
Two angles that can be put together to form a straight line — like $\cl"red"{∠BAC}$ and $\cl"red"{∠CAD}$ — are called supplementary angles. Each row of the table below gives a measure for $\cl"red"{∠BAC}$. By sliding the slider to that value, give the measure of the supplementary angle $\cl"red"{∠CAD}$, as well as the sum of the two measures.
As you can see:
The measures of any two supplementary angles add up to 180˚.
This makes sense, because you can think of a straight line as forming a 180˚ angle.
An acute angle is smaller than its supplement (so its measure is less than 90˚). An obtuse angle is larger than its supplement (so its measure is greater than 90˚). A right angle is equal in measure to its supplement (so its measure is 90˚). In the figure to the left, $\cl"red"{∠BAC}$ and $\cl"red"{∠CAD}$ are right angles, while $\cl"blue"{∠FEG}$ is an acute angle and $\cl"blue"{∠GEH}$ is an obtuse angle. Note that “acute” means sharp, and “obtuse” means blunt.
For each row of the table below, click the Next button to see two supplementary angles $\cl"red"{∠BAC}$ and $\cl"red"{∠CAD}$. Say whether each angle is acute, obtuse, or right.
The figure to the left shows two lines which intersect in a point. You can slide the slider below the figure to move one of the lines by changing the angles between the lines.
Let’s say that the measure of $\cl"red"{∠BAC}$ (in degrees) is $x$. Notice that $\cl"red"{∠BAC}$ and $\cl"red"{∠EAB}$ are supplementary angles, because they can be put together to form the straight line $\cl"red"{CE}$. Since the measure of supplementary angles must add up to 180˚, this means that the measure of $\cl"red"{∠EAB}$ (in degrees) must be $180 - x$.
In the last question, you saw that if $\cl"red"{∠BAC}$ has measure $x$ (in degrees), then $\cl"red"{∠DAE}$ has measure $180-(180-x)$ (also in degrees). Simplify this formula.
Two angles which are on the opposite sides of intersecting lines — like $\cl"red"{∠BAC}$ and $\cl"red"{∠DAE}$ — are called vertical angles. As you have seen:
Vertical angles have equal measures.
The figure to the left shows two parallel red lines $AC$ and $\cl"red"{DF}$, along with a blue line $BE$ that intersects both of them. The slider allows you to change the measure of the angle $\cl"purple"{∠FEB}$ between the blue line and the bottom red line.
A line which intersects two other lines in different points (the way the blue line intersects the two red lines) is called a transversal. Two angles like $\cl"purple"{∠FEB}$ and $\cl"purple"{∠CBG}$ which are formed by a transversal and each of the two lines it intersects, and that are in the same position relative to those lines, are called corresponding angles.
The picture for transversal-qn showed you two parallel lines intersected by a transversal, and suggested that corresponding angles are equal whenever this happens. Now we’ll study why this is true, by learning some more about how translations relate to parallel lines.
The figure to the left shows a red line $AB$ and a blue arrow. The $\cl"blue"\direction$ slider rotates the blue arrow, and the $\cl"gray"\distance$ slider translates the line in the direction that the blue arrow is pointing. The translated line $\cl"green"{A'B'}$ is drawn in green.
The direction of the red line is 45˚ away from pointing to the right. If you set the $\cl"blue"\direction$ slider to $45˚$, the blue arrow will point in the same direction as the red line, moving from $\cl"red"A$ to $\cl"red"B$.
If you set the $\cl"blue"\direction$ slider to $-135˚$, the blue arrow will still point in the same direction as the red line, only now moving from $\cl"red"B$ to $\cl"red"A$.
In general:
The translation of a line either coincides with the original line (when you translate in the direction of the line) or is parallel to the original line (when you translate in any other direction).
The figure to the left shows two lines $\cl"red"{AC}$ and $\cl"red"{DF}$, intersected by a transversal $\cl"blue"{BE}$. The corresponding angles $\cl"purple"{∠CBG}$ and $\cl"purple"{∠FEB}$ are equal in measure (they both measure 60˚). The slider translates the lines $\cl"red"{DF}$ and $\cl"blue"{BE}$ in the direction of (along) the line $\cl"blue"{BE}$, while leaving the line $\cl"red"{AC}$ alone. The translated line $\cl"green"{D'F'}$ is drawn in green.
Because $\cl"purple"{∠F'E'B'}$ is a translation of $\cl"purple"{∠FEB}$, they have the same measure. So the line $\cl"green"{D'F'}$ always makes an angle of 60˚ with the line $\cl"blue"{BE}$.
Whenever you have two lines intersected by a transversal, and a pair of corresponding angles which are equal in measure, you can translate one of the lines so it coincides with the other in this way. This means that:
When two lines are intersected by a transversal and there is a pair of corresponding angles that have equal measures, the lines are parallel.
The figure to the left shows two parallel lines $\cl"red"{AB}$ and $\cl"red"{CD}$. The sliders allow you to translate the line $\cl"red"{AB}$. The translated line $\cl"green"{A'B'}$ is drawn in green.
If you translate a line (like $\cl"red"{AB}$) so that any one point on it coincides with a point on a parallel line (like $\cl"red"{CD}$), it will make the entire line coincide with the parallel line.
The figure to the left shows two parallel lines $\cl"red"{AC}$ and $\cl"red"{DF}$, intersected by a transversal $\cl"blue"{BE}$. The slider translates the lines $\cl"red"{DF}$ and $\cl"blue"{BE}$ in the direction of (along) the line $\cl"blue"{BE}$, while leaving the line $\cl"red"{AC}$ alone. The translated line $\cl"green"{D'F'}$ is drawn in green. Let’s apply the results of translation-qn and parallel-postulate-qn to this situation, and see if that explains what you saw in transversal-qn.
In summary, when two parallel lines are intersected by a transversal, the corresponding angles can be connected by a translation, and therefore have equal measures. This is what transversal-qn suggested:
When two parallel lines are intersected by a transversal, the corresponding angles have equal measures.
The figure to the left shows two parallel red lines $AC$ and $\cl"red"{DF}$, along with a blue transversal $BE$.
Click to see another figure with two parallel red lines $AC$ and $\cl"red"{DF}$, along with a blue transversal $BE$ intersecting them at a different angle. Given that the measure of $\cl"purple"{∠FEB}$ is 65˚, fill in the measure of each other angle in the table below.