--- title: "Finding angles with parallel lines" subtitle: "Angles and shapes · Parallel lines · Lesson 2" --- # Finding angles with parallel lines **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A carpenter measures one angle where a brace crosses two parallel shelves. Without measuring again, she knows every other angle. How? :::teacher-note **In one sentence:** Students use the parallel-line rules (corresponding and alternate equal, co-interior add to 180°) with straight-line and vertically opposite angles to find unknown angles from one given angle, naming a reason for each (AC9M7M04, node 2). **Students should be able to say:** I can state the rules: with parallel lines, corresponding and alternate angles are equal and co-interior angles add to 180°. I can find an unknown angle using one of these rules, and say which rule. I can find angles in two steps, giving a reason for each step. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Corresponding and alternate angles are equal. Is a co-interior pair the same? :::keep-together **Two parallel lines are crossed by a transversal. $c = 65^\circ$, and $e$ is co-interior with $c$. What is $e$?** ☐ $65^\circ$ ☐ $295^\circ$ ☐ $25^\circ$ ☐ $115^\circ$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 115°. Co-interior angles are not equal: they add to 180°, so e = 180° − 65° = 115°. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## From one angle to all eight :::keep-together ### Equal pairs Here is how one angle gives all eight, with a reason for each. 1. $b = 70^\circ$ — Given. 2. $c = 70^\circ$ — Vertically opposite $b$: equal. 3. $a = 110^\circ$ — On a straight line with $b$: $180 - 70$. 4. $f = 70^\circ$ — Corresponding to $b$: equal. 5. $e = 110^\circ$ — Co-interior with $c$: they add to $180^\circ$. 6. $g = 70^\circ,\; h = 110^\circ$ — Corresponding to $c$ and to $d$ (or vertically opposite $f$ and $e$). Then find $f$ in the picture. **Find $f$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 72 b and f are corresponding angles, so f = 72°. ::: :::keep-together ### Find the mistake **Leo finds $f$ when $d = 110^\circ$. Click the line where his working goes wrong.** ``` 1 d = 110° (given). 2 d and f are between the parallel lines, on the same side. 3 So d and f are co-interior angles. 4 Co-interior angles are equal, so f = 110°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: co-interior angles add to 180°, so f = 180° − 110° = 70°. Only corresponding and alternate angles are equal. ::: :::keep-together ### Co-interior **Find $f$ correctly.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 70 Co-interior angles add to 180°: f = 180° − 110° = 70°. ::: :::typical ## One change at a time **Alternate.** Each task changes one thing. Give the reason as well as the size. **Co-interior from below.** **Two steps.** **Check a chain.** **Not parallel.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Angle and reason :::keep-together ### Corresponding **Find $c$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 80 Corresponding angles are equal: c = 80°. ::: :::keep-together ### Co-interior **Find $d$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 118 Co-interior angles add to 180°: d = 180° − 62° = 118°. ::: :::keep-together ### The reason **$e = 120^\circ$ and $c = 60^\circ$. Which reason links them?** ☐ Alternate angles are equal ☐ Co-interior angles add to 180° ☐ Corresponding angles are equal ::: :::teacher-note **Answer:** Co-interior angles add to 180° c and e are between the lines on the same side, and 60° + 120° = 180°. ::: :::keep-together ### Review: names **Review: which pair is alternate?** ☐ $d$ and $e$ ☐ $a$ and $d$ ☐ $d$ and $h$ ☐ $d$ and $f$ ::: :::teacher-note **Answer:** $d$ and $e$ d and e: between the lines, on opposite sides. ::: ## Make: equal co-interior angles Then: can alternate angles ever add to $180^\circ$? When? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A transversal crosses two parallel lines. One angle is 75°. Find the angle corresponding to it, and the angle next to that one on a straight line, with a reason for each.** {{lines-3}} ::: :::teacher-note **Answer:** 75° (corresponding angles are equal), then 180° − 75° = 105° (angles on a straight line). :::