Understanding Alternate Interior Angles Theorem and Theorems About Lines and Angles

Angle Addition Postulate	Isolate $m ∠ 2$
$m ∠ 1 + m ∠ 2$ $=$ $18 0^{\circ}$	$m ∠ 2$ $=$ $18 0^{\circ} - m ∠ 1$
$m ∠ 2 + m ∠ 3$ $=$ $18 0^{\circ}$	$m ∠ 2$ $=$ $18 0^{\circ} - m ∠ 3$

Angle Addition Postulate

Isolate

m ∠ 2

m ∠ 1 + m ∠ 2

=

18 0^{\circ}

m ∠ 2

=

18 0^{\circ} - m ∠ 1

m ∠ 2 + m ∠ 3

=

18 0^{\circ}

m ∠ 2

=

18 0^{\circ} - m ∠ 3

$x = 19$
$m ∠ 1 = 3 x + 14$	$m ∠ 2 = 6 x - 5$
$m ∠ 1 = 3 (19) + 14$	$m ∠ 2 = 6 (19) - 5$
$m ∠ 1 = 57 + 14$	$m ∠ 2 = 114 - 5$
$m ∠ 1 = 7 1^{\circ}$	$m ∠ 2 = 10 9^{\circ}$

x = 19

m ∠ 1 = 3 x + 14

m ∠ 2 = 6 x - 5

m ∠ 1 = 3 (19) + 14

m ∠ 2 = 6 (19) - 5

m ∠ 1 = 57 + 14

m ∠ 2 = 114 - 5

m ∠ 1 = 7 1^{\circ}

m ∠ 2 = 10 9^{\circ}

Converse Corresponding Angles Theorem
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Converse Corresponding Angles Theorem
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Reflection Across $C M$
Preimage	Image
$B$	$A$
$C$	$C$
$M$	$M$

Reflection Across

C M

Preimage

Image

B

A

C

C

M

M

Solve for $x .$

a diagram showing a pair of corresponding angles

a diagram showing a pair of alternate interior angles and a pair of corresponding angles

Examining the diagram, we see that our angles have the same position in relation to the transversal and their respective lines, so they are corresponding angles. Also, since the two lines cut by the transversal are parallel, we know by the Corresponding Angles Theorem that the angles are congruent.

Let's solve this equation for x by performing inverse operations.

To make things more clear, we will color the expressions for the given angle measures.

We have three relationships to consider. Angles:& 10x-10^(∘) & 3x+32^(∘) [-0.2em] Relationship:& Vertical angles [1em] Angles:& 10x-10^(∘) & 12x-20^(∘) [-0.2em] Relationship:& Alternate interior angles [1em] Angles:& 12x-20^(∘) & 3x+32^(∘) [-0.2em] Relationship:& Corresponding angles Vertical angles are always congruent. This is also true for alternate interior angle and corresponding angles if the two lines cut by the transversal are parallel. In this case, we do not have this information. Therefore, we can only claim that the vertical angles are congruent. 10x-10^(∘)=3x+32^(∘) Let's solve this equation for x by performing inverse operations.

Determine the measure of $α .$

a diagram with a pair of parallel lines cut by two transversals

Let's mark a pair of corresponding angles in the diagram. Since the two lines cut by the transversal that create the corresponding angles are parallel, we know by the Corresponding Angles Theorem that they are congruent angles.

Now we can find the adjacent angles to the right of the 35^(∘)-angle in the original diagram.

Notice that α and the 75^(∘)-angle are vertical angles. By the Vertical Angles Theorem, we know that they are congruent. Therefore, we have that α=75^(∘).

Find $x$ from the following diagram.

A diagram with two pairs of parallel lines and a pair of angle measures given

To relate the angles, we will introduce a third angle ∠ θ. For clarity purposes, we will also color code the given angle measures.

Notice that both of our expressions represent corresponding angles to ∠ θ. Since the two lines cut by the transversal are parallel, we know that they are congruent by the Corresponding Angles Theorem. Let's identify both pairs of corresponding angles.

Since both angles are congruent to ∠ θ, we know by the Transitive Property of Congruence that they are themselves congruent. m∠ θ = 15x-30^(∘) m∠ θ = 10x+5^(∘) ⇓ 10x+5^(∘) = 15x-30^(∘) Let's solve this equation using inverse operations.

Find the length of $\overline{A B} .$

A perpendicular bisector DC that intersects AB at point D

A perpendicular bisector DB that intersects AC at E

The Converse of the Perpendicular Bisector Theorem states that when a point is equidistant from the endpoints of a segment, then it is on the segment's perpendicular bisector. From the diagram, we see that DC is in fact the perpendicular bisector of AB. Therefore, AD and DB must be congruent segments.

With this information, we can write the following equation. AD= DB ⇓ 6x= 4x+4 Let's perform inverse operations to find the value of x.

Finally, to find the length of AB, we will substitute 2 for x in the expression 5x and multiply by 2 to account for AD and DB.

From the diagram, we see that E is the midpoint of AC and that DB is the perpendicular bisector to AC through E. By the Perpendicular Bisector Theorem, we conclude that AB= BC.

Since AB= BC, we can write an equation to find x by substituting the expressions for AB and BC. AB= BC ⇓ 10x+1= 7x+13 Let's solve the equation for x.

We can now determine AB by substituting 4 for x in the corresponding expression.

Kriz says that $∠ X V W$ is congruent to $∠ R W S .$ Is there enough information to confirm what Kriz says? Explain.

Since both ∠ XVW and ∠ RWS are below their respective lines and to the left of the transversal, they must be corresponding angles.

Corresponding angles are congruent by the Corresponding Angles Theorem if the two lines cut by the transversal are parallel. Even though the lines appear to be parallel, we have not been given this information explicitly.

Therefore, we cannot conclude that ∠ XVW and ∠ RWS have equal measures and, therefore, are congruent. Kriz is not necessarily correct.

Find the length of $\overline{H G} .$

a perpendicular bisector to a segment GJ

a perpendicular bisector to a segment HJ

Examining the diagram, we can see that K is the midpoint of GJ. We also see that HK is perpendicular to GJ through K. Therefore, HK must be the perpendicular bisector to GJ.

According to the Perpendicular Bisector Theorem, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Since H is a point on the perpendicular bisector, we can conclude that HG and HJ are congruent segments. HG≅ HJ ⇒ HG=GJ Let's show this on the diagram.

Since HJ is 4 units long, we conclude that HG is also 4 units long.

In this exercise, we are given in the diagram that KH=KJ. According to the Converse of the Perpendicular Bisector Theorem, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. This tells us that K lies on the perpendicular bisector of HJ.

Notice that G also lies on the perpendicular bisector. Therefore, it is also equidistant from the endpoints of HJ. Since we know the length of GJ, we can also determine the length of HG.

Therefore, HG=1 unit.

Statements	Reasons
$ℓ_{1}$ and $ℓ_{2}$ lines	Given
$∠ 1$ and $∠ 2$ supplementary	Definition of straight angle
$m ∠ 1 + m ∠ 2 = 18 0^{\circ}$	Definition of supplementary angles
$m ∠ 2 = 18 0^{\circ} - m ∠ 1$	Subtraction Property of Equality
$∠ 2$ and $∠ 3$ supplementary	Definition of straight angle
$m ∠ 2 + m ∠ 3 = 18 0^{\circ}$	Definition of supplementary angles
$m ∠ 2 = 18 0^{\circ} - m ∠ 3$	Subtraction Property of Equality
$18 0^{\circ} - m ∠ 1 = 18 0^{\circ} - m ∠ 3$	Transitive Property of Equality
$m ∠ 1 = m ∠ 3$	Subtraction and Multiplication Properties of Equality

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

Statement	Reason
$∠ 1 ≅ ∠ 2$	Given
$∠ 2 ≅ ∠ α$	Vertical Angles Theorem
$∠ 1 ≅ ∠ α$	Transitive Property of Congruence
$ℓ_{1} ∥ ℓ_{2}$	Converse Corresponding Angles Theorem

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

Statement	Reason
$∠ 1 ≅ ∠ 2$	Given
$∠ 2 ≅ ∠ α$	Vertical Angles Theorem
$∠ 1 ≅ ∠ α$	Transitive Property of Congruence
$ℓ_{1} ∥ ℓ_{2}$	Converse Corresponding Angles Theorem

	13 Theory slides
	12 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Statements	Reasons
$\overline{A M} ≅ \overline{M B}$ $∠ A M C$ and $∠ B M C$ are right angles	Definition of a perpendicular bisector.
$∠ A M C ≅ ∠ B M C$	All right angles are congruent.
$\overline{C M} ≅ \overline{C M}$	Reflexive Property of Congruence.
$△ A C M ≅ △ B C M$	SAS Congruence Theorem.
$\overline{A C} ≅ \overline{B C}$	Corresponding parts of congruent figures are congruent.
$A C = B C$	Definition of congruent segments.

Theorems About Lines and Angles

Catch-Up and Review

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Theorems About Lines and Angles

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