Reasoning · Grade 4-2 Parallel Lines and Angles
Problem
Give the four working points names.
Angles are named by three points, so nothing can be written down until every corner in the picture has a name.
4.G.A.1Draw A DiagramTurning the 150 gives a neighbour of 30 degrees.
Two angles that sit side by side on a straight line always fill up 180° between them, so knowing one instantly gives the other.
7.G.B.5Identify SubproblemsIn the right triangle the second angle is 60 degrees.
In a right triangle the right angle uses up half of the 180° budget, so the two slanted corners have to share the remaining 90° — 30° at one end forces 60° at the other.
8.G.A.5Identify SubproblemsThe parallel sides let the 130 be carried down.
One straight line cutting two parallel lines makes the same pattern of angles at both crossings, so a measure known on the top side is automatically known on the bottom side.
8.G.A.5Identify SubproblemsThe 130 degrees found at the top side is carried straight down to the bottom side, because the rectangle's two long sides are parallel.
One line crossing two parallel lines makes matching angles at both crossings, so a measure known at one is known at the other.
Once the angle is on the bottom side, it can be added to or taken from its neighbours there, since angle measures combine like lengths.
Turning it around gives 50 degrees.
Straight lines again: the segment leans one way, and whatever it leaves on the right it takes from the left, so the two angles at Q must total 180°.
7.G.B.5Identify SubproblemsIn the small triangle the crossing angle is 100 degrees.
Both known angles now sit at the two ends of the same base line, and the triangle's 180° budget hands the rest to the crossing point.
8.G.A.5Identify SubproblemsFlipping to the other side gives 80 degrees.
Ray XR and ray XD are the two halves of one straight segment through X, so the angle on one side of it and the angle on the other side always fill 180° together.
7.G.B.5Identify SubproblemsParallel sides carry an angle across the figure for free, and every triangle you find hands you its last corner for 180° minus the other two.