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Before any theorems will be introduced, try to discover some properties of angles using the interactive applet. While exploring, think about how those properties could be proven. The applet allows for translations and rotations of the angles. Consider a pair of vertical angles.
Vertical angles can be mapped onto each other by using a rotation. Since rotations are rigid motions, the angle measures are preserved. This leads to the following theorem.
Vertical angles are always congruent.
Based on the characteristics of the diagram, the following relations hold true.
$∠1≅∠3$
$∠2≅∠4$
Analyzing the diagram, it can be seen that $∠1$ and $∠2$ form a straight angle, so these are supplementary angles. Similarly, $∠2$ and $∠3$ are also supplementary angles.
Therefore, by the Angle Addition Postulate, the sum of $m∠1$ and $m∠2$ is $180_{∘},$ and the sum of $m∠2$ and $m∠3$ is also $180_{∘}.$ These facts can be used to express $m∠2$ in terms of $m∠1$ and in terms of $m∠3.$
Angle Addition Postulate | Isolate $m∠2$ |
---|---|
$m∠1+m∠2$ $=$ $180_{∘}$ | $m∠2$ $=$ $180_{∘}−m∠1$ |
$m∠2+m∠3$ $=$ $180_{∘}$ | $m∠2$ $=$ $180_{∘}−m∠3$ |
$LHS−180_{∘}=RHS−180_{∘}$
$LHS⋅(-1)=RHS⋅(-1)$
The previous proof can be summarized in the following two-column table.
Statements | Reasons |
$ℓ_{1}$ and $ℓ_{2}$ lines | Given |
$∠1$ and $∠2$ supplementary | Definition of straight angle |
$m∠1+m∠2=180_{∘}$ | Definition of supplementary angles |
$m∠2=180_{∘}−m∠1$ | Subtraction Property of Equality |
$∠2$ and $∠3$ supplementary | Definition of straight angle |
$m∠2+m∠3=180_{∘}$ | Definition of supplementary angles |
$m∠2=180_{∘}−m∠3$ | Subtraction Property of Equality |
$180_{∘}−m∠1=180_{∘}−m∠3$ | Transitive Property of Equality |
$m∠1=m∠3$ | Subtraction and Multiplication Properties of Equality |
Consider the points $A,$ $B,$ $C,$ and $D$ on each ray that starts at the point of intersection $E$ of the two lines.
Suppose that points $A$ and $B$ are rotated $180_{∘}$ about point $E.$Use the given expressions to form an equation for $x.$ Identify the relationship between $∠1$ and $∠3,$ as well as $∠2$ and $∠4$ by analyzing their positions.
$x=19$ | |
---|---|
$m∠1=3x+14$ | $m∠2=6x−5$ |
$m∠1=3(19)+14$ | $m∠2=6(19)−5$ |
$m∠1=57+14$ | $m∠2=114−5$ |
$m∠1=71_{∘}$ | $m∠2=109_{∘}$ |
Consider two parallel lines cut by a transversal. The applet shows a pair of corresponding angles, $∠A$ and $∠B.$ Is it possible to translate one line so that one of these angles maps onto another?
The observed relation between corresponding angles is presented and proven in the following theorem.
If $ℓ_{1}∥ℓ_{2},$ then $∠1≅∠5,$ $∠2≅∠6,$ $∠3≅∠7,$ and $∠4≅∠8.$
Note that the converse statement is also true.
If $∠1≅∠5,$ $∠2≅∠6,$ $∠3≅∠7,$ or $∠4≅∠8,$ then $ℓ_{1}∥ℓ_{2}.$
This theorem can be proven by an indirect proof. Let $ℓ_{1}$ and $ℓ_{2}$ be two lines intersected by a transversal line $ℓ_{3}$ forming corresponding congruent angles $∠1$ and $∠2.$
Since the goal is to prove that $ℓ_{1}$ is parallel to $ℓ_{2}$, it will be temporarily assumed that $ℓ_{1}$ and $ℓ_{2}$ are not parallel.In a Flowerland Village house, there are stairs with hand railings like shown in the diagram. The measures of $∠1$ and $∠2$ are expressed as $5t−2$ and $4t+12,$ respectively.
What are the measures of $∠1$ and $∠2?$How do measures of $∠1$ and $∠2$ relate to each other? Use the given expressions to form an equation for $t.$
Like corresponding angles, alternate interior angles are also formed by two parallel lines cut by a transversal.
If $ℓ_{1}∥ℓ_{2},$ then $∠1≅∠2$ and $∠3≅∠4.$
To prove that alternate interior angles are congruent, it will be shown that $∠1$ and $∠2$ are congruent.
Notice that by definition $∠2$ and $∠5$ are vertical angles. By the Vertical Angles Theorem, they are therefore congruent angles.The previous proof can be summarized in the following two-column table.
Statements | Reasons |
$∠2$ and $∠5$ are vertical angles | Def. of vertical angles |
$∠2≅∠5$ | Vertical Angles Theorem |
$∠5$ and $∠1$ are corresponding angles | Def. of corresponding angles |
$∠5≅∠1$ | Corresponding Angles Theorem |
$∠2≅∠1$ | Transitive Property of Congruence |
Apart from the points of intersection, consider two more points on each line.
Next, $A,$ $B,$ $C,$ and $D$ will be translated parallel to the transversal until the points $A,$ $C,$ and $D$ lie on $ℓ_{2}.$ Then, $A,$ $B,$ and $C$ will be rotated $180_{∘}$ about $F.$ It should be noted that since point $D$ lies on the transversal, when translating it to $ℓ_{2}$ the point will fall into the same position as $F.$ Therefore, $D$ will not be affected by the rotation around $F.$The converse statement is also true.
If $∠1≅∠2$ or $∠3≅∠4,$ then $ℓ_{1}∥ℓ_{2}.$
The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate interior angles and one more angle.
It needs to be proven that $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. It is already given that $∠1$ is congruent to $∠2.$
Converse Corresponding Angles Theorem |
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. |
Since $∠1$ and $∠α$ are corresponding congruent angles, then $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. To summarize, all of the steps will be described in a two-column proof.
Statement | Reason |
$∠1≅∠2$ | Given |
$∠2≅∠α$ | Vertical Angles Theorem |
$∠1≅∠α$ | Transitive Property of Congruence |
$ℓ_{1}∥ℓ_{2}$ | Converse Corresponding Angles Theorem |
Similar properties can be discovered for alternate exterior angles.
If $ℓ_{1}∥ℓ_{2},$ then $∠1≅∠2$ and $∠3≅∠4.$
In order to prove that alternate exterior angles are congruent, it will be shown that $∠1$ and $∠2$ are congruent.
Notice that by definition, $∠2$ and $∠8$ are corresponding angles. Therefore, by the Corresponding Angles Theorem, they are congruent angles.The previous proof can be summarized in the following two-column table.
Statements | Reasons |
$∠2$ and $∠8$ are corresponding angles | Def. of corresponding angles |
$∠2≅∠8$ | Corresponding Angles Theorem |
$∠8$ and $∠1$ are vertical angles | Def. of vertical angles |
$∠8≅∠1$ | Vertical Angles Theorem |
$∠2≅∠1$ | Transitive Property of Congruence |
Consider the points of intersection as well as two more points on each line.
Next, $A,$ $B,$ $C,$ and $D$ will be translated in the direction of the transversal so that points $A,$ $C,$ and $D$ lie on $ℓ_{2}.$ Then, $A,$ $B,$ and $C$ will be rotated $180_{∘}$ about $F.$$∠ADB≅∠GFH$ and $∠CDB≅∠EFH$
If $∠1≅∠2$ or $∠3≅∠4,$ then $ℓ_{1}∥ℓ_{2}.$
The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate exterior angles and one more angle.
It needs to be proven that $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. It is already given that $∠1$ is congruent to $∠2.$
Converse Corresponding Angles Theorem |
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel. |
Since $∠1$ and $∠α$ are corresponding congruent angles, $ℓ_{1}$ and $ℓ_{2}$ are parallel lines. Each step of the proof will now be summarized in a two-column proof.
Statement | Reason |
$∠1≅∠2$ | Given |
$∠2≅∠α$ | Vertical Angles Theorem |
$∠1≅∠α$ | Transitive Property of Congruence |
$ℓ_{1}∥ℓ_{2}$ | Converse Corresponding Angles Theorem |
In order to build Tulip Street on the south side of the Lilian river, which goes through Flowerland Village, there is a need to build a bridge. Devontay, an architect, proposed the following plan for the bridge.
It is known that the measure of $∠1$ is equal to $4a+11$ and the measure of $∠2$ is equal to $8a−53.$ What are the measures of $∠1$ and $∠2?$
How do the measures of $∠1$ and $∠2$ relate to each other? Use the given expressions to form an equation for $a.$
By analyzing the diagram it can be noted that $∠1$ and $∠2$ are alternate interior angles.