--- title: "Reasoning with angles" subtitle: "Angles and shapes · Parallel lines · Lesson 3" --- # Reasoning with angles **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A detective does not guess: every step has evidence. Finding an angle is the same. Each angle you find needs a reason, and the reasons are what make the answer trustworthy. :::teacher-note **In one sentence:** Students find unknown angles at parallel lines in two-step chains, with a reason for each step, writing their working their own way; the platform marks the answer and the reason at the key step, then shows a model to compare with (AC9M7M04, node 3). **Students should be able to say:** I can find an unknown angle in two or more steps, using parallel-line, straight-line and vertically opposite angles. I can give a reason for each step, naming the rule. I can use angles to show whether two lines are parallel. **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 $b$ and $h$ are not one of the pairs. Which angle could you find first? :::keep-together **Two parallel lines are crossed by a transversal. $b = 70^\circ$. What is $h$?** ☐ $20^\circ$ ☐ $70^\circ$ ☐ $110^\circ$ ☐ $250^\circ$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 110°. b and h are not a named pair, so it takes two steps: f = 70° (corresponding to b), then h = 180° − 70° = 110° (f and h are on a straight line). ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Reasons for every step :::keep-together ### Know, choose, say why Here is the routine for every step: what you know, the rule that links it to a new angle, and the reason. 1. $b = 70^\circ$ — Know: the given angle. 2. $f = 70^\circ$ — Choose: $f$ is in the same place as $b$ at the other crossing. Say why: corresponding angles are equal. 3. $h = 180^\circ - 70^\circ$ — Choose: $h$ is next to $f$ on the bottom line. Say why: angles on a straight line add to $180^\circ$. 4. $h = 110^\circ$ — Done: two steps, two reasons. Now find $h$ yourself. Write it your own way; the checker marks your answer and the reason for the first step, then shows you a model to compare. **Find $h$. Write each step with its reason.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $f = 70^\circ$ (corresponding to $b$: corresponding angles are equal); $h = 180^\circ - 70^\circ = 110^\circ$ ($f$ and $h$ are on a straight line) f = 70° (corresponding to b), h = 110° (a straight line with f). Another route: d = 110° (a straight line with b), h = 110° (corresponding to d). ::: :::keep-together ### Find the mistake **Kai finds $e$ when $b = 70^\circ$. Click the line where his reasoning goes wrong.** ``` 1 b = 70° (given). 2 b and e look like a pair, so b and e are alternate angles. 3 Alternate angles are equal, so e = 70°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: b and e are not a named pair (b is above the top line, e above the bottom line, on different sides). Looking related is not a reason. e = 110°: f = 70° (corresponding to b), then e = 180° − 70° (a straight line). ::: :::keep-together ### Another chain **Find $g$. Write each step with its reason.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $e = 115^\circ$ (alternate to $d$: alternate angles are equal); $g = 180^\circ - 115^\circ = 65^\circ$ ($e$ and $g$ are on a straight line) e = 115° (alternate to d), g = 65° (a straight line with e). Or: f = 65° (co-interior with d), g = 65° (vertically opposite f). ::: :::typical ## One change at a time **A different start.** Each task changes one thing. Your working is yours; the marked parts are the answer and one reason. **Up from the bottom.** **Your own route.** **Not parallel.** **Showing lines are parallel.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Angle chains :::keep-together ### Down the page **Find $a$, with a reason for each step.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $c = 75^\circ$ (corresponding to $g$); $a = 180^\circ - 75^\circ = 105^\circ$ ($a$ and $c$ are on a straight line) c = 75° (corresponding to g), a = 105° (a straight line with c). ::: :::keep-together ### Through the middle **Find $h$, with a reason for each step.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $e = 180^\circ - 80^\circ = 100^\circ$ (co-interior with $c$); $h = 100^\circ$ (vertically opposite $e$) e = 100° (co-interior with c), h = 100° (vertically opposite e). Or g = 80° (corresponding to c), h = 100° (a straight line with g). ::: :::keep-together ### Review: one crossing **Review: Tom says "$b$ and $c$ are alternate angles". Which is right?** ☐ They are vertically opposite: both at the top crossing ☐ They are corresponding ☐ Tom is right ::: :::teacher-note **Answer:** They are vertically opposite: both at the top crossing b and c are at the same crossing, opposite each other: vertically opposite (and equal for that reason). ::: ## Make: set it up Write your two steps first, then turn the line. Stretch: write your own two-step puzzle for a partner: give one angle, ask for one that is not a named pair with it, and write the model reasons. 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, making a 120° angle. Find the angle alternate to it and the angle co-interior to it, with reasons.** {{lines-3}} ::: :::teacher-note **Answer:** 120° (alternate angles are equal) and 180° − 120° = 60° (co-interior angles add to 180°). :::