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29 Congruent triangles GCSE Mathematics for OCR (Foundation) In this chapter you will learn how to … • prove that two triangles are congruent using the cases SSS, ASA, SAS, RHS. • apply congruency in calculations and simple proofs. or more resources relating F to this chapter, visit GCSE Mathematics Online. Using mathematics: real-life applications Congruent triangles are used in construction to reinforce structures that need to be strong and stable. “When designing any bridge I have to allow for reinforcement. This ensures that the bridge doesn’t collapse under heavy traffic. Any bridge I design has (Structural engineer) many congruent triangles.” Before you start … Ch 5 You need to know how to label angles and shapes that are equal. C 1 Here are two identical triangles. D a Write down a pair of sides A that are equal in length. b What angle is equal in size B to angle BAC? c Write down another pair of F angles that are equal in size. Ch 9 Ch 5, 9 Ch 5 You need to know basic angle facts. 2 Match up the correct E A statement with the correct diagram. a Vertically opposite angles are equal. b Alternate angles are equal. c Corresponding angles are equal. C B You should be able to apply angle facts to find angles in figures and to justify results in simple proofs. 3 Decide whether each statement is true You need to know and be able to apply the properties of triangles and quadrilaterals. 4 What is the value of x? Choose the correct answer. A B or false. a Angle DBE 5 40° (alternate to angle ADB). bAngle BEC 5 50° (complementary to 40° angle ADB). c Angle BDE 5 angle BED 5 70°. D dTriangle ABD, triangle BDE and triangle BCE are congruent. E A D x A60° B30° C 45° P D50° B 450 C C © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms 29 Congruent triangles Assess your starting point using the Launchpad Step 1 1 Identify which pairs of triangles are congruent. Give reasons for your decisions. a Go to Section 1: Congruent triangles 6 cm 8 cm 8 cm 6 cm b 9 mm 4 mm 6 mm 4 mm 6 mm 9 mm c 2m 25° 80° 25° 75° 2m ✓ Step 2 2 Quadrilateral ABCD is a kite. A a Prove that triangle ACD is congruent to triangle ACB. b Prove that angle ADC 5 angle ABC D M B Go to Section 2: Applying congruency C ✓ Go to Chapter review Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers 451 GCSE Mathematics for OCR (Foundation) Key vocabulary congruent: shapes that are identical in shape and size. Section 1: Congruent triangles Congruent triangles are identical in shape and all corresponding measurements are equal. The corresponding sides are equal in length. The corresponding angles are the same size. x x A B y Tip If you place two congruent triangles on top of each other the angles and sides will match up. The matching sides and angles are the corresponding sides or angles. y z z Congruent triangles can have different orientations. When the triangles are in different orientations you need to think carefully about the corresponding sides and angles. This diagram shows triangle A and B from the example above in different orientations. x x A B y z y z Two triangles are congruent if one of the following sets of conditions is true. Side side Side or SSS: the three sides of one triangle are equal in length to the three sides of the other triangle. Angle side angle or ASA: two angles and one side of one triangle are equal to the corresponding two angles and side of another triangle. D A B C F E F D A C B E Key vocabulary included angle: the angle between two lines that meet at a vertex. Side angle side or SAS: two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. Right angle hypotenuse side or RHS: the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one other side of the other right-angled triangle. 452 D A F B C A B E D C F E © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms 29 Congruent triangles Tip It is important to write the letters of the vertices of the two triangles in the correct order. When we write that triangle ABC is congruent to triangle DEF, it means that: Â 5 D̂ B E B̂ 5 Ê Ĉ 5 F̂ and AB 5 DE A C D F AC 5 DF and BC 5 EF. The conditions in the table are the minimum conditions for proving that triangles are congruent. No other combinations of side and angle facts are sufficient to tell us whether a triangle is congruent or not. For example: Two triangles with all their angles equal can still be very different sizes. FF DD AA CC BB EE If you are given two triangles that have two equal sides and one equal angle, but where the equal angle is not included (between the two given sides), you do not know if the triangles are congruent or not. The third side might have a different length in the two triangles. G 6 cm6 cm J G 7 cm7 cm J 7 cm7 cm 6 cm6 cm 40° 40° H H 40° 40° I I K K L L Although a pair of triangles with one of these sets of information could still be congruent, the conditions given are not sufficient to prove that they are. Tip If two congruent shapes are drawn in different orientations it is sometimes hard to see which angles and sides match each other. To help you, trace one shape onto tracing paper and label its vertices, then rotate and/or flip the paper over to help see which sides and angles match up. Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers 453 GCSE Mathematics for OCR (Foundation) Work it out 29.1 Here are three proofs for congruence for the pair of triangles. P X 85° 85° 5 cm Which one uses the correct reasoning? Why are the others incorrect? 5 cm 30° 65° R 6 cm Q Y 6 cm Z Option A Option B Option C In triangle PQR and triangle XYZ: In triangle PRQ and triangle XYZ: In triangle PRQ and triangle XYZ: PR 5 XY 5 5 cm PR 5 XY 5 5 cm angle P 5 angle X 5 85° In triangle PRQ, angle Q 5 65° (sum of angles in a triangle) RQ 5 YZ 5 6 cm angle Q 5 angle Z 5 65° so triangle PQR is congruent to triangle XYZ (SAS). RQ 5 YZ RQ 5 YZ 5 6 cm In triangle XYZ, angle Y 5 30° (sum of angles in a triangle) so the triangles are congruent. so triangle PRQ is congruent to triangle XYZ (SAS). Exercise 29A 1 Match up each of the congruency descriptions (SSS, ASA, SAS, RHS) with each pair of triangles below: b a A E C F K H J B D L c t s E d s t H E D u u G F 2 Which of the following figures show a pair of congruent triangles? In your answer, state whether the triangles are congruent or not, or whether there is insufficient information. Write the triangles with the vertices in the correct order andDgive the reasons as SSS, SAS, ASA or RHS. J J 5 cm5 cmL D3 cm 3 cm a a E E b b G G A A a a 3 cm3 cm A B3 cm B3 cm B B D D3 cm3 cm E A3 cm3 cm 3 cm3 cm 3 cm3 cm 3 cm3 cm C C C CF F F c c c c P P M M M 100° M 100° N 10° 100° 10° 100° P 5 cm5 cm N 10° 10° P 5 cm5 cm 454 b b 4 cm4 cmG 4 cm4 cm H H H H F R R Q Q 10° 10° Q100° Q100° R R 10° 10° 100°100° 5 cm5 cm N 5 cm5 cm N S S S E S d dT d dT G L 20°5 cm 20° J J 5 cm L 4 cm4 cm 5 cm5 cm 20° 20° L 20°5 cm 20°5 cm 4 cm4 cm I I 20° 20° K K I I K K T W W T 20° 20°8 cm8 cm W W 8 cm8 cm 20° 20° 8 cm8 cm 20° 20° 20° 20°8 cm8 cm U U V VY Y X X U U V VY Y X X © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms 29 Congruent triangles 3 Prove that triangle ABC is congruent B D to triangle DEC. C A E 4 Write down two different proofs for congruence of triangles DEF and DGF. D E 9m F 9m 5 In the diagram, PQ is parallel to SR and G R P QT 5 TR 5 2 cm. 2 cm Prove that triangle PQT is congruent to triangle SRT. T 2 cm 50° S Q 6 Prove that triangles ABE and CBD in the figure are congruent, giving full reasons. A B E C D 7 Triangle ABD is isosceles. AC is the perpendicular height. Prove that triangle ABC is congruent to triangle ADC. A B C Tip What do we know about line AC in an isosceles triangle? D Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers 455 GCSE Mathematics for OCR (Foundation) 8 In the figure below, PR 5 SU and RTUQ is a kite. Prove that triangle PQR is congruent to triangle SQU. P U T Q R S 9 ABCD in the figure is a kite. A D B E C Prove that: a triangle ADB is congruent to triangle CDB. b triangle AED is congruent to triangle CED. 10 Quadrilateral ABCD is a rhombus. A D E B C Prove that: a triangle AED is congruent to triangle CEB. b triangle AEB is congruent to triangle CED. Section 2: Applying congruency Whenever you learn new skills and concepts in geometry, you add them to your toolbox and use them in problem-solving. So you will need to combine what you’ve learnt previously with your new skills to solve problems. The steps in the following framework are useful for solving geometry problems. 456 © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms 29 Congruent triangles Problem-solving framework In the diagram, AM 5 BM and PM 5 QM. Q A Tip a Prove that triangle AMP is congruent to triangle BMQ. You saw in Chapter 5 that the symbol // means ‘is parallel to’. M b Prove that AP // QB. B P Steps for approaching a problem‑solving question What you would do for this example Step 1: Read the question carefully to decide what you have to find. Find mathematical evidence to show that the triangle AMP is congruent to triangle BMQ; look for SSS, SAS, RHS or ASA. Find evidence to show that lines AP and QB are parallel. Step 2: Write down any further information that might be useful. Lines AB and PQ intersect, so the two triangles also have vertically opposite angles. Vertically opposite angles are equal (see Chapter 9 if you need to). Step 3: Decide what method you’ll use. We are given two equal sides and we can see that the included angle is also equal, so use SAS to prove congruence. Step 4: Set out your working clearly. a In triangles AMP and BMQ: AM 5 BM (given). PM 5 QM (given). angle AMP 5 angle BMQ (vertically opposite angles at M). So triangle AMP is congruent to triangle BMQ (SAS). b Angle APM 5 BQM (matching angles of congruent triangles). So AP // QB (alternate angles are equal). Worked example 1 Triangle DEF is divided by GF into two smaller triangles. Prove that FG is perpendicular to DE in the diagram. E G D In triangle FGD and triangle FGE: Side FG is common to both triangles. F DG 5 EG (given) DF 5 EF (given) So the triangles are congruent (SSS). Angle DGF 5 angle EGF and the two angles lie on a straight line (angles on a straight line add up to 180°). So each angle 5 90°, and FG is perpendicular to DE. Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers 457 GCSE Mathematics for OCR (Foundation) Exercise 29B 1 In the diagram, prove that KL 5 ML. J K L M 2 Use the facts given in the A D diagram to: a prove that angle E ABE 5 angle EDC. b prove that quadrilateral B ABCD is a parallelogram. C 3 In the quadrilateral, SP 5 SR and P S QP // RS. Angle QRP 5 56°. a Calculate the size of angle PQR and give reasons. b Find the size of angle PSR and 56° give reasons. Q R 4 In the diagram, PQ 5 PT , QR 5 ST and angle PQR 5 angle PTS. Prove that triangle PRS is isosceles. P Q R S T 5 In the figure below, prove that: a triangle AEB is congruent to triangle CEB. b angle EAD 5 angle ECD. B E A 458 D C © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms 29 Congruent triangles 6 In quadrilateral ABCD, AD 5 BC and AD // BC. Prove that angle ABC 5 angle ADC. A D 1 1 B C Checklist of learning and understanding Congruent triangles You can prove that two triangles are congruent using one of the four cases of congruence: ○Side side side or SSS: the three sides of one triangle are equal in length to the three sides of the other triangle. ○Angle side angle or ASA: two angles and one side of one triangle are equal to the corresponding two angles and side of another triangle. ○Side angle side or SAS: two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. ○Right angle hypotenuse Side or RHS: the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one other side of the other right-angled triangle. You can apply your knowledge of congruency to help prove other geometrical features such as parallel and perpendicular lines. Chapter review 1 State whether each pair of triangles is congruent or not. Give a reason for or additional questions on F the topics in this chapter, visit GCSE Mathematics Online. your answer and give the vertices of the triangles in the correct order. a D A b H B c 4 cm C F 3 cm E d Q A 7 cm B 4 cm R 8 cm I K J H 3 cm 25° L 8 cm 30° G 3 cm 7 cm 5 cm 30° 5 cm 4 cm 3 cm J G 3 cm 3 cm L 25° I K C S Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers 459 GCSE Mathematics for OCR (Foundation) 2 In the figure below, prove that angle T 5 angle R. T Q P S R 3 Show that angle EBC 5 angle ECB in the figure below. A B C D E 4 In the figure below, triangle ABD lies between two parallel lines. BC 5 CA 5 AD. Prove that angle EAD 5 2 × angle ABC. A B C E D 5 Prove that WX 5 XV in the diagram. W X V Y 460 © Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms