--- title: "Angles in a triangle" subtitle: "Angles and shapes · Angle sums · Lesson 4" --- # Angles in a triangle **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A roof truss, a bike frame, a folded paper plane: triangles of every shape. One thing about their angles never changes. :::teacher-note **In one sentence:** Students drag triangles of every shape on a dot grid, see that the three angles always add to 180°, and explain why by tearing off the corners to make a straight line (AC9M7M05, node 1). **Students should be able to say:** I can show that the angles of a triangle add to 180°, by measuring and by tearing off the corners. I can explain why this is true for every triangle, whatever its shape or size. I can tell whether three angles could be the angles of a triangle. **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 Picture a triangle you could pull out into a long, thin shape. Which angles change? :::keep-together **You drag one corner of a triangle so one angle gets much bigger. What happens to the total of the three angles?** ☐ It gets bigger ☐ It gets smaller ☐ It stays the same ☐ It depends on how long the sides are My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** It stays the same: 180°. As one angle grows, the other two shrink by exactly as much. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Why 180°? :::keep-together ### Find the mistake **Kai compares a small triangle with one twice as big. Click the line where his reasoning goes wrong.** ``` 1 Kai draws a small triangle, then one with every side twice as long. 2 Small triangle: 50° + 60° + 70° = 180°. 3 The big triangle has longer sides, so its angles are bigger. 4 So the big triangle's angles add to more than 180°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: making every side twice as long does not change the angles, only the size. The big triangle's angles are still 50°, 60° and 70°, adding to 180°. ::: :::keep-together ### Why a straight line? **Why do the three torn corners always make a straight line?** ☐ Because every triangle has a $90°$ angle ☐ Because the angles add to $180°$, and a straight line is $180°$ ☐ Because the corners are all the same size ::: :::teacher-note **Answer:** Because the angles add to $180°$, and a straight line is $180°$ Side by side at one point, the three angles fill a straight line exactly. Angles on a straight line add to 180°, so the triangle's angles do too. ::: :::typical ## One change at a time **A right angle.** Each task changes one thing. First: make a square corner. **Very flat.** Now make one angle very big. **Three equal angles.** **Two right angles?.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Use the rule :::keep-together ### Possible? **Which three angles could be the angles of a triangle?** ☐ $50°, 60°, 80°$ ☐ $90°, 90°, 10°$ ☐ $40°, 60°, 80°$ ::: :::teacher-note **Answer:** $40°, 60°, 80°$ 40° + 60° + 80° = 180°. ::: :::keep-together ### Measuring by hand **Lia measures a paper triangle with a protractor: $57°, 68°, 56°$. That adds to $181°$. What is the best explanation?** ☐ She has measured the wrong angles ☐ Her triangle is a special one that adds to $181°$ ☐ Her measurements are a little out: a protractor reads to about a degree ::: :::teacher-note **Answer:** Her measurements are a little out: a protractor reads to about a degree Real measurements are rounded. 181° is within a degree or two of 180°, which is what measuring three angles by hand gives. ::: :::keep-together ### Bigger triangle **Triangle P has sides 3 cm, 4 cm and 5 cm. Triangle Q has sides 30 cm, 40 cm and 50 cm. How do their angle sums compare?** ☐ Q's is 10 times bigger ☐ They are the same: $180°$ ☐ Q's is bigger, but not 10 times ::: :::teacher-note **Answer:** They are the same: $180°$ Every triangle's angles add to 180°. Q is P made 10 times bigger, with exactly the same angles. ::: :::keep-together ### Review **Review: two angles sit side by side on a straight line. One is $130°$. How big is the other?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 50 180° − 130° = 50°. ::: ## Make: one angle equals the other two Think first: if one angle is the other two added together, what fraction of $180°$ is it? Then build it. Stretch: can a triangle have one angle that is exactly twice the sum of the other two? How big is it? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A student says the angles of a very large triangle add to more than 180°. Explain why they are wrong.** {{lines-3}} ::: :::teacher-note **Answer:** Every triangle's angles add to 180°: tear off the corners and they fill a straight line. Size changes the sides, not the angles. :::