8th Grade · 8.G.A.5

Parallel lines cut by a transversal in 8th grade

Parallel lines cut by a transversal make eight angles, and many of them are equal. Parallel lines run side by side and never meet. A transversal is a third line that crosses both of them. In 8th grade, your child uses these angle facts to explain why the angles of every triangle add up to $180^\circ$ . The same facts show when two triangles are similar.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

What it means, in plain words

Where the transversal crosses each parallel line, it makes four angles. Angles in the same position at the two crossings are called corresponding angles, and they are equal. Alternate interior angles sit between the parallel lines on opposite sides of the transversal, and they are equal too. Same-side interior angles sit between the parallel lines on the same side of the transversal, and they add up to $180^\circ$ . Two angles that together make a straight line add up to $180^\circ$ . From one known angle, your child can find all eight.

The same facts explain the angle sum of a triangle. Draw a line through one corner of a triangle, parallel to the opposite side. Along that line, three angles sit side by side and make $180^\circ$ . The middle one is the triangle's own angle at that corner. The two outer ones are alternate interior angles, so they equal the triangle's other two angles. The three angles of any triangle therefore add up to $180^\circ$ .

Exterior angles come next. An exterior angle is the angle outside a triangle, made by extending one side past a corner. It equals the two inside angles that are not next to it, added together. Children also meet the angle-angle rule: if two angles of one triangle equal two angles of another, the triangles are similar, meaning they have the same shape. Work on parallel lines cut by a transversal in 8th grade is mostly explaining facts like these with a picture, not writing a formal proof.

Your child should be able to:

Name corresponding angles and alternate interior angles and use the fact that they are equal
Find all eight angles from one angle when parallel lines are cut by a transversal
Explain with a picture why the angles of a triangle add up to $180^\circ$
Find an exterior angle of a triangle and use two equal angles to show that two triangles are similar

Example

Two parallel lines are cut by a transversal. One of the eight angles is $65^\circ$ . What are the other seven?

The angle beside it on the same straight line is $180^\circ - 65^\circ = 115^\circ$ .
Angles directly across from each other at a crossing are equal, so this crossing has two $65^\circ$ angles and two $115^\circ$ angles.
Corresponding angles are equal, so the other crossing has the same four angles.
Three of the other angles are $65^\circ$ and four are $115^\circ$ .

Example

A triangle has angles of $50^\circ$ and $70^\circ$ . What is the third angle, and what is the exterior angle at that third corner?

The three angles add up to $180^\circ$ : $180^\circ - 50^\circ - 70^\circ = 60^\circ$ .
The exterior angle makes a straight line with the $60^\circ$ angle: $180^\circ - 60^\circ = 120^\circ$ .
It also equals the two other inside angles added together: $50^\circ + 70^\circ = 120^\circ$ .
The third angle is $60^\circ$ , and the exterior angle at that corner is $120^\circ$ .

Where children usually get stuck

Your child marks two angles as equal when they actually add up to $180^\circ$ , and writes $x = 113^\circ$ where the answer is $67^\circ$ .
Equal angles get so much attention that every pair starts to look equal. A narrow angle (less than $90^\circ$ ) and a wide angle (more than $90^\circ$ ) can never be equal.
What helps. Before any arithmetic, ask whether each angle looks narrow or wide. In a diagram with parallel lines, a narrow angle and a wide angle are never equal, so they add up to $180^\circ$ .
Your child uses the equal-angle facts on two lines that are not parallel.
The facts look as if they work for any two lines crossed by a third. They only hold for parallel lines, which a diagram marks with small matching arrows.
What helps. Ask your child to point to the arrow marks or the word "parallel" before using any of the equal-angle facts.
For an exterior angle, your child adds all three inside angles, or adds the inside angle right next to it.
The rule is remembered as "add the angles" without which angles. The exterior angle equals the two inside angles far from it, not the one beside it.
What helps. Draw a triangle, extend one side past a corner with a ruler and cut out a paper copy of the triangle. Tear off the two corners that are not at the exterior angle. Laid side by side inside the exterior angle, they fill it exactly.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

Complete the reasoning below to compare the two triangles.

Third angle of triangle PQR	$\displaystyle m\angle R = 180^\circ - 40^\circ - 65^\circ =$ $\displaystyle ^\circ$
Angle relationship	$\displaystyle \text{Relationship of } \angle R \text{ and } \angle U \text{: }$
Conclusion	$\displaystyle \text{Conclusion for } \triangle PQR \text{ and } \triangle STU \text{: }$

Show the answer and how to solve it

The blank is 75, the relationship is equal, and the conclusion is similar because both triangles have angle measures $40^{\circ}$ , $65^{\circ}$ , and $75^{\circ}$ .

1The two triangles have matching angle pairs in the same positions: angle P and angle S are both $40^{\circ}$ (blue), and angle Q and angle T are both $65^{\circ}$ (green). The unknown angles are at vertex R in triangle PQR and at vertex U in triangle STU.
2Use the triangle angle sum to find angle R. The same calculation applies in triangle STU, giving angle U = $75^{\circ}$ , so angle R and angle U are equal.
3All three angle pairs match: $40^{\circ}$ , $65^{\circ}$ , and $75^{\circ}$ . The matching angles give the triangles the same shape. One triangle can be enlarged or reduced to match the other, so they are similar. This is angle-angle (AA) similarity.

Common mistake. A common mistake is to add $40^{\circ}$ and $65^{\circ}$ and stop at $105^{\circ}$ , or to write congruent for the triangles. You must subtract from $180^{\circ}$ to get $75^{\circ}$ , and matching angles prove the triangles are similar, not necessarily congruent.

The idea underneath. If two angle pairs match, the third pair also matches because each triangle totals $180^{\circ}$ . Matching angles fix the same triangle shape, even when the sizes differ. One triangle can be scaled to match the other, so the triangles are similar. This is angle-angle (AA) similarity.

2On level

Look at the diagram. Sofia's argument for why the angles in triangle ABC add to

180^{\circ}

is shown below. Which statement best describes Sofia's argument?

Step 1	$\displaystyle m\angle 1 = m\angle A \text{ because } \overline{DE} \parallel \overline{AB}$
Step 2	$\displaystyle m\angle 3 = m\angle B \text{ because } \overline{DE} \parallel \overline{AB}$
Step 3	$\displaystyle m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \text{ because they lie on a straight line}$
Conclusion	$\displaystyle m\angle A + m\angle B + m\angle C = 180^\circ$

The premises are reversed. Angle 1 corresponds to angle B, not angle A, and angle 3 corresponds to angle A, not angle B.

Parallel lines create supplementary pairs with the base angles here, not equal pairs. Angles 1 and 3 cannot be set equal to the base angles.

There is no mistake. The argument is correct.

The argument never justifies why angles 1, 2, and 3 add to

180^{\circ}

. That reason about the straight line is missing.

Show the answer and how to solve it

Every step is correct: the correspondences follow from DE $\parallel$ AB, the three top angles form a straight angle of $180^{\circ}$ , and the substitution is valid. There is no mistake.

1Look at the diagram. Line DE passes through vertex C and is marked parallel to AB with matching arrows. Angles 1, 2, and 3 sit side by side along the straight line DE.
2Because DE is parallel to AB, angle 1 matches angle A and angle 3 matches angle B (alternate interior angles). These equal-angle steps are correct.
3Angles 1, 2, and 3 are adjacent along the straight line DE, so they form a straight angle and add to $180^{\circ}$ . The stated reason (they lie on a straight line) is exactly right.
4Now check Sofia's work. The correspondences are right, the parallel reason is stated, the straight-line total is $180^{\circ}$ , and substituting gives $m\angle A + m\angle B + m\angle C = 180^{\circ}$ . Every step is correct.
5Because no step contains an error, the correct choice is that there is no mistake. The argument is a complete, valid proof that the triangle's angles sum to $180^{\circ}$ .

Common mistake. A common mistake is to assume every 'find the flaw' argument must contain a flaw and pick a plausible-sounding error. Here each step is justified, so the right answer is that nothing is wrong.

The idea underneath. A helper-line proof replaces two of the triangle's angles with equal angles at C, so the three triangle angles become three adjacent angles on one straight line that add to $180^{\circ}$ .

3On level

Use the diagram to complete the proof. Type only the missing angle labels or the number 180.

Straight line	$\displaystyle \angle a + \angle Q + \angle b =$ $\displaystyle ^\circ$
Left match	$\displaystyle \angle P = \angle$
Right match	$\displaystyle \angle R = \angle$
Conclusion	$\displaystyle \angle P + \angle Q + \angle R = 180^\circ$

Show the answer and how to solve it

The blanks are 180, a, and b. This completes the proof that $\angle P + \angle Q + \angle R = 180^{\circ}$ .

1Look at the helper line through Q. It has a parallel marker matching the base PR, so the helper line is parallel to PR.
2Because the lines are parallel, the left base angle matches the left helper angle, and the right base angle matches the right helper angle.
3Angles a, Q, and b lie along a straight line at vertex Q, so they add to $180^{\circ}$ .
4Substitute $\angle P$ for $\angle a$ and $\angle R$ for $\angle b$ to conclude the triangle's interior angles sum to $180^{\circ}$ .

Common mistake. A common mistake is swapping labels a and b — writing $\angle P = \angle b$ or $\angle R = \angle a$ . Match the left base angle to the left helper angle (P matches a) and the right base angle to the right helper angle (R matches b). Also remember that a straight line measures $180^{\circ}$ , not $360^{\circ}$ .

The idea underneath. A line through a triangle vertex parallel to the opposite side turns the triangle's three interior angles into three adjacent angles on one straight line.

4Stretch
Look at the diagram and Priya's work for x. Which statement best describes the work?
$\displaystyle 113 + x$
$\displaystyle =$
$\displaystyle 180$
$\displaystyle x$
$\displaystyle =$
$\displaystyle 77$
A
The mistake is that these two marked angles should be equal, so adding them to $180^{\circ}$ is the wrong setup.
B
The mistake is that these angles should add to $180^{\circ}$ , not have the same measure.
C
The mistake is arithmetic: the angle relationship is correct, but the subtraction is wrong.
D
The mistake is that the student found the angle next to x, not x itself.
E
There is no mistake; the work is correct.
Show the answer and how to solve it
The correct statement is that the mistake is arithmetic: the angle relationship is correct, but the subtraction is wrong. These same-side interior angles are supplementary, so $x = 67°$ .
1. 1The two marked angles are between the parallel lines and on the same side of the transversal, so they are same-side interior angles.
2. 2Same-side interior angles are supplementary, so their measures add to $180°$ . The setup $113 + x = 180$ is correct.
3. 3Priya's first line is correct. The angle relationship is not the problem.
4. 4Subtracting 113 from 180 gives $x = 67°$ , not $77°$ . There is a subtraction slip of 10.
5. 5The mistake is arithmetic. The correct value is $67°$ , not $77°$ .
Common mistake. A common mistake is checking only the setup and trusting the arithmetic without verifying it. Even with the correct equation $113 + x = 180$ , the subtraction $180 - 113$ must be checked carefully; the answer is 67, not 77.
The idea underneath. First decide whether the marked angles are equal or supplementary; only after that does the arithmetic matter.

$\displaystyle 113 + x$	$\displaystyle =$	$\displaystyle 180$
$\displaystyle x$	$\displaystyle =$	$\displaystyle 77$