--- title: "Unknown angles in shapes" subtitle: "Angles and shapes · Angle sums · Lesson 6" --- # Unknown angles in shapes **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A roof truss, a kite, a bridge brace: builders work out the angle they cannot measure from the ones they can. Every step needs a reason. :::teacher-note **In one sentence:** Students find unknown angles in triangles, quadrilaterals and pentagons in steps, with a reason for each (angle sum, straight line, isosceles), and explain why an exterior angle equals the sum of the two interior opposite angles; working is their own, the answer and a key reason are marked (AC9M7M05, node 3). **Students should be able to say:** I can find an unknown angle in a triangle or polygon using its angle sum. I can combine angle sums with straight-line and parallel-line angles, giving a reason for each step. I can explain why an exterior angle of a triangle equals the sum of the two interior opposite angles. **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 Two steps: which angle can you find first? :::keep-together **A triangle has $a = 50^\circ$ and $b = 60^\circ$. The side through $C$ is extended past $C$, making an outside angle $x$ with the third side. What is $x$?** ☐ $70^\circ$ ☐ $110^\circ$ ☐ $120^\circ$ ☐ $130^\circ$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 110°. c = 180° − 50° − 60° = 70° (angles in a triangle), then x = 180° − 70° = 110° (a straight line). Notice: 110° = 50° + 60°. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Angle sums in steps :::keep-together ### Triangle, then a straight line Here is the routine: know, choose the rule, say why. 1. $a = 59^\circ,\; b = 45^\circ$ — Know: the given angles. 2. $c = 180^\circ - 59^\circ - 45^\circ = 76^\circ$ — Choose: $a$, $b$, $c$ are a triangle's angles. Why: angles in a triangle add to $180^\circ$. 3. $x = 180^\circ - 76^\circ = 104^\circ$ — Choose: $c$ and $x$ sit on one straight line. Why: angles on a straight line add to $180^\circ$. 4. $x = 59^\circ + 45^\circ$ — A shortcut you can now explain: an exterior angle equals the sum of the two interior opposite angles. Now find $x$ yourself, your own way. **Find $x$. Write each step with its reason.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $c = 180^\circ - 59^\circ - 45^\circ = 76^\circ$ (angles in a triangle add to $180^\circ$); $x = 180^\circ - 76^\circ = 104^\circ$ ($c$ and $x$ are on a straight line) c = 76° (angles in a triangle), x = 104° (a straight line with c). Check: 59° + 45° = 104°. ::: :::keep-together ### Find the mistake **Zoe finds the fourth angle of a quadrilateral. Click the line where her working goes wrong.** ``` 1 a = 76°, b = 68°, c = 102° (given). 2 The angles of a shape add to 180°. 3 d = 180° − 76° − 68° − 102°. 4 d = −66°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: a quadrilateral's angles add to (4 − 2) × 180° = 360°, not 180°. d = 360° − 76° − 68° − 102° = 114°. A negative angle on line 4 is the clue that something earlier went wrong. ::: :::keep-together ### A quadrilateral **Find $d$, with your reason.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $a + b + c + d = 360^\circ$ (angles in a quadrilateral: $(4 - 2) \times 180^\circ$); $d = 360^\circ - 76^\circ - 68^\circ - 102^\circ = 114^\circ$ (subtract the angles you know) (4 − 2) × 180° = 360°, so d = 360° − 246° = 114°. ::: :::typical ## One change at a time **Isosceles.** Each task changes one thing. **A pentagon.** **Exterior at another corner.** **A right angle.** **Impossible triangles.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Unknown angles :::keep-together ### Triangle **A triangle has angles of $35^\circ$ and $85^\circ$. Find the third angle.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 60 180° − 35° − 85° = 60°. ::: :::keep-together ### Quadrilateral **Three angles of a quadrilateral are $90^\circ$, $85^\circ$ and $120^\circ$. Find the fourth.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 65 360° − 295° = 65°. ::: :::keep-together ### Exterior angle backwards **A triangle has an exterior angle of $130^\circ$ at $C$, and $a = 70^\circ$. Find $b$, with your reasons.** Answer: {{field-15}} ::: :::teacher-note **Answer:** $c = 180^\circ - 130^\circ = 50^\circ$ ($c$ and the exterior angle are on a straight line); $b = 180^\circ - 70^\circ - 50^\circ = 60^\circ$ (angles in a triangle add to $180^\circ$) c = 50°, b = 60°. Or: 130° = a + b, so b = 60°. ::: :::keep-together ### Review: parallel lines **Review: parallel lines, $c = 70^\circ$. Which is a correct first step to find $e$?** ☐ $e = 70^\circ$: alternate to $c$ ☐ $e = 70^\circ$: corresponding to $c$ ☐ $e = 110^\circ$: co-interior with $c$ ::: :::teacher-note **Answer:** $e = 110^\circ$: co-interior with $c$ Co-interior angles add to 180°: e = 110°. ::: ## Make: a given exterior angle First work out what $c$ must be. Then find a triangle that has it. Stretch: design an angle puzzle for a partner with two unknown angles, 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 triangle has angles of 48° and 67°. Find the third angle, and the exterior angle next to it, with reasons.** {{lines-3}} ::: :::teacher-note **Answer:** 180° − 48° − 67° = 65° (angles in a triangle). Exterior angle 180° − 65° = 115° (a straight line). Check: 48° + 67° = 115°. :::