--- title: "Congruent triangles" subtitle: "Angles and shapes · Congruence and similarity · Lesson 16" --- # Congruent triangles **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A builder cuts a hundred identical roof trusses. To check two are the same, does she need to measure every side and every angle, or are three measurements enough? :::teacher-note **In one sentence:** Students build triangles from given parts and see that SSS, SAS, ASA, AAS and RHS fix one triangle while SSA can give two (AC9M8SP01, node 1). **Students should be able to say:** I can name the four congruence tests (SSS, SAS, ASA or AAS, RHS). I can choose the test that shows two triangles are congruent. I can explain why SSA is not a congruence test. **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 You do not need all six. These **congruence tests** each fix a triangle, so two triangles that share them are congruent. The angle between two sides is the **included angle**. | Test | Equal parts | |---|---| | **SSS** | all three sides | | **SAS** | two sides and the included angle (the angle **between** them) | | **ASA** / **AAS** | two angles and a matching side | | **RHS** | a right angle, the hypotenuse and one other side | :::keep-together **Two triangles both have sides of 8 cm and 6 cm and a 40° angle. Must they be congruent?** ☐ Yes: two sides and an angle are enough ☐ No: you always need all three sides ☐ Not always: it depends where the angle is My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** Not always. With the angle between the two sides (SAS) the triangle is fixed. With the angle somewhere else (SSA), two different triangles can fit. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Which test :::keep-together ### Which test Here is the routine. 1. $\text{AB} = 6,\ \text{BC} = 4,\ \angle B = 70^\circ$ — List the equal parts. 2. $\angle B \text{ is between AB and BC}$ — Where is the angle? 3. $\text{SAS}$ — Two sides and the included angle. 4. $\triangle ABC \equiv \triangle PQR$ — Write the congruence with matching corners in order. **In both triangles, AB = 5 cm, AC = 7 cm and angle A = 40°. Which test shows they are congruent?** ☐ SSS ☐ ASA ☐ They might not be congruent ☐ SAS: angle A is between AB and AC ::: :::teacher-note **Answer:** SAS: angle A is between AB and AC Angle A is where AB and AC meet: two sides and the included angle. ::: :::keep-together ### Find the mistake **Tom decides two triangles are congruent. Click the line where his reasoning goes wrong.** ``` 1 Two triangles each have a 6 cm side, a 5 cm side and a 40° angle. 2 In both, the 40° angle is opposite the 5 cm side, not between the two sides. 3 Two sides and an angle match, so they are congruent by SAS. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: SAS needs the angle between the two sides. Here it is not (SSA), and SSA can give two different triangles. ::: :::keep-together ### Angles only **Two triangles both have angles of 50°, 60° and 70°. Are they congruent?** ☐ Yes, by ASA ☐ Yes: all three angles match ☐ Not necessarily: they are the same shape but can be different sizes ☐ No, never ::: :::teacher-note **Answer:** Not necessarily: they are the same shape but can be different sizes Equal angles make them similar (same shape). A matching side is needed for congruence. ::: :::typical ## Tests that fail **No triangle.** **RHS.** **Sides that fail.** **Matching order.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Congruence tests :::keep-together ### SSS **Both triangles have sides 4, 6 and 7 cm. Which test?** ☐ SSS ☐ SAS ☐ RHS ☐ Not enough to tell ::: :::teacher-note **Answer:** SSS Three equal sides: SSS. ::: :::keep-together ### AAS **Both triangles have angles 30° and 80°, and the side opposite the 80° angle is 9 cm. Which test?** ☐ SSS ☐ AAS ☐ SAS ☐ Not enough to tell ::: :::teacher-note **Answer:** AAS Two angles and a side that is not between them: AAS. ::: :::keep-together ### SSA **Both triangles have sides 9 cm and 6 cm, and a 35° angle opposite the 6 cm side (not between the two sides). Must they be congruent?** ☐ Yes, by RHS ☐ Yes, by SSS ☐ No: SSA is not a congruence test ☐ Yes, by SAS ::: :::teacher-note **Answer:** No: SSA is not a congruence test Two sides and a non-included angle can make two different triangles. ::: :::keep-together ### Review: angle sum **Review: a triangle has angles of 47° and 68°. What is the third angle, in degrees?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 65 180 − 47 − 68 = 65°. ::: ## Make: exactly one Then explain why SSA still is not a congruence test. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Two triangles have AB = PQ = 7 cm, BC = QR = 5 cm and angle B = angle Q = 60°. Are they congruent? Which test?** {{lines-3}} ::: :::teacher-note **Answer:** Yes, by SAS: angle B is between AB and BC. :::