Problem

Draw an auxiliary parallel line to find a bent angle

Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ . Find the measure of angle $\bigcirc$ .

Segment $\overline{AB}$ lies horizontally on the left. At point $B$ the path bends downward to point $C$ , and the interior angle at $B$ measures $120^\circ$ . At point $C$ the path bends back upward to point $D$ , and the interior angle at that vertex is $\bigcirc$ . From point $D$ a horizontal segment $\overline{DE}$ extends to the right, and the interior angle at $D$ measures $110^\circ$ . Segment $\overline{AB}$ is parallel to segment $\overline{DE}$ .

Geometry

Show worked solution try it yourself first ✦

1 · Understandwhat's really being asked

A path goes from A to B (horizontal), bends down to C, bends back up to D, then runs horizontally to E. The top segment AB is parallel to the top segment DE. The interior angle at B is 120 deg and at D is 110 deg. I need the interior angle at C (marked with a circle).

Givens

Segment AB is parallel to segment DE.
The interior angle at B is 120 deg.
The interior angle at D is 110 deg.
AB is horizontal on the left, DE is horizontal on the right, and C is the low point between them.

Unknowns

The measure of the angle at C (the circle).

Constraints

Angles on a straight line, or co-interior angles between parallel lines, are used.
All four points form one continuous zigzag path with AB and DE parallel.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw an auxiliary line through C that is parallel to both AB and DE. This splits the angle at C into two pieces, each a co-interior (same-side) angle with one of the parallel segments. Each piece is then 180 deg minus the given angle, and adding them gives the full angle at C.

3 · Execute4 carry out the plan

1Draw an auxiliary parallel line through C

#1 Draw a Diagram 4.G.A.1

Through C, draw a horizontal line parallel to AB and DE. The angle at C is split into a left part (between CB and the new line) and a right part (between the new line and CD).

\angle C = \angle(\text{left part}) + \angle(\text{right part})

Sliding a parallel line through the corner lets each side of the bend talk to a line it is parallel to, turning one hard angle into two friendly ones.

2Left part using AB

#7 Identify Subproblems 4.MD.C.7

BC crosses the parallel lines AB and the auxiliary line. The interior angle at B is 120 deg, so the co-interior angle (left part at C) on the same side is 180 deg - 120 deg.

180^\circ - 120^\circ = 60^\circ

Between two parallel lines, the two same-side angles add to a straight 180 deg, so the partner of 120 deg is 60 deg.

🧠 Why? · Feynman

Where segment BC crosses parallel line AB and the auxiliary line through C, the left part of the bend at C measures 180 degrees minus the 120 degree angle at B, which is 60 degrees.

Why?

The 120 degree angle at B and the left part of the angle at C are same-side interior angles made where BC cuts the two parallel lines, and a same-side pair always fills a straight 180 degrees together, so the left part is what is left after taking 120 from 180.

Why?

Line BC cuts across both parallel lines, and a line crossing two parallels makes the matching corner angles equal, so the angle BC opens at C in that matching corner is the same 120 degrees it opens at B.

🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.

Why?

That matching 120 degree angle at C and the left part of the bend sit side by side along the straight auxiliary line through C, so the two of them together make one straight angle.

🧱A straight angle is 180 degreesAngles sitting side by side along a straight line always fill up 180 degrees.

3Right part using DE

#7 Identify Subproblems 4.MD.C.7

CD crosses the auxiliary line and DE. The interior angle at D is 110 deg, so the co-interior angle (right part at C) is 180 deg - 110 deg.

180^\circ - 110^\circ = 70^\circ

Same idea on the right: the same-side partner of 110 deg is 70 deg.

4Add the two parts

#7 Identify Subproblems 4.MD.C.7

The angle at C is the sum of its left and right parts.

60^\circ + 70^\circ = 130^\circ

Angle measures simply add when two angles sit side by side around the same vertex.

Answer: 130 degrees

4 · Reviewdoes it hold up?

C is a wide, downward-opening bend, so an obtuse answer above 90 deg fits the picture. The two pieces 60 deg and 70 deg are each less than the straight 180 deg they came from, and their sum 130 deg is a believable reflex-free angle.

Another way: Use Identify Subproblems with a triangle: extend BC and DC ideas to form a triangle whose angles are 60 deg, 70 deg, and the exterior turn, and confirm the interior angle at C is 130 deg via the angle sum.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Drawing the auxiliary line through C parallel to AB and DE.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Finding 180-120 and 180-110, then adding 60 and 70 to get the angle at C.

💡Takeaway. Slide a parallel line through the bend and the tricky angle splits into two easy 'straight-line leftovers' you just add up!

▶ Practice — 10 problems