Perpendicular is half of a straight angle

4.G.A.14.MD.C.7 · take · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

Segment $\overline{CO}$ and segment $\overline{MO}$ are perpendicular to each other. Find the measure of angle $a$ .

[Figure] A horizontal line $CD$ is drawn through point $O$ (left end $C$ , right end $D$ ). From point $O$ , segment $\overline{MO}$ is drawn straight up, perpendicular to segment $\overline{CO}$ . A second straight line $EF$ passes through point $O$ , with $E$ at the upper left and $F$ at the lower right. The angle between ray $\overline{OE}$ and the upward segment $\overline{MO}$ is marked as $70^\circ$ . The angle $a$ is the angle between ray $\overline{OD}$ (horizontal, to the right) and ray $\overline{OF}$ (down to the right).

Show solution

Understand

A horizontal line CD passes through point O. Segment MO points straight up from O and is perpendicular to CO. A second straight line EF also passes through O, with E up-left and F down-right. The angle between ray OE and the upward segment MO is 70 deg. I need angle a, between ray OD (horizontal, right) and ray OF (down-right).

Givens

CD is a straight horizontal line through O.
MO is perpendicular to CO, so angle MOD = 90 deg and angle MOC = 90 deg.
EF is a straight line through O (E up-left, F down-right), so OE and OF are opposite rays.
The angle MOE between MO (up) and OE (up-left) is 70 deg.
Angle a is between OD (right) and OF (down-right).

Unknowns

The measure of angle a (angle DOF).

Constraints

Angles on one side of a straight line at a point add to 180 deg.
Vertical (opposite) angles are equal.
Perpendicular lines make a 90 deg angle.

Plan

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

Use the angles around O on the upper-left side. The straight horizontal ray OC, the upward ray OM (90 deg from it), and the ray OE lie around O. The angle MOE is 70 deg, so angle EOM plus angle MOD makes the angle from OE around to OD. Angle a (between OD and OF) is the vertical angle to angle EOC (or found directly from the straight-line total), giving a quick subtraction.

Execute

#1 Draw a Diagram 4.G.A.2

MO is straight up and perpendicular to the horizontal line, so angle MOD = 90 deg on the right and angle MOC = 90 deg on the left. Ray OE is 70 deg from MO toward the left, inside the upper-left region.

\angle MOE = 70^\circ, \quad \angle MOC = 90^\circ

Perpendicular means a clean 90 deg between the upward segment and the horizontal line, a fixed reference to measure OE from.

#7 Identify Subproblems 4.MD.C.7

On the upper-left, the 90 deg angle MOC is split by ray OE into angle MOE (70 deg) and angle EOC. So angle EOC is the leftover.

\angle EOC = 90^\circ - 70^\circ = 20^\circ

The upward segment to OE uses 70 deg of the 90 deg quarter-turn, leaving 20 deg from OE down to the horizontal line OC.

#7 Identify Subproblems 4.MD.C.7

OE and OF are opposite rays, and OC and OD are opposite rays, so angle DOF (which is a) is the vertical angle of angle EOC and therefore equal to it.

a = \angle DOF = \angle EOC = 20^\circ

Crossing lines make equal angles straight across, so the small 20 deg angle on the upper-left reappears as a on the lower-right.

Answer: 20 degrees

Review

OE is only 20 deg above the horizontal line on the left, so its opposite ray OF dips just 20 deg below the horizontal on the right, making angle a a small acute angle, which matches the gentle downward slant of OF in the figure.

Use the straight line MF idea (tool 7): angle MOE = 70 deg and angle MOD = 90 deg give angle EOD = 160 deg; since EOD and DOF are supplementary on line EF, a = 180 - 160 = 20 deg.

Standards · min grade 4

4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the perpendicular MO to fix the 90 deg reference angles at O.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Computing 90 - 70 = 20 deg and matching it across as the vertical angle a.

💡 Take 70 deg out of the 90 deg right angle to get 20 deg, then bounce it straight across the crossing: angle a is 20 deg!