Similarity and Congruence

Introduction

This lesson discusses two important geometry ideas: similarity and congruence. Both are used to compare figures based on shape, size, side relationships, and angle relationships.

Basic Competencies

1. Explain and determine similarity and congruence among plane figures.
2. Solve problems related to similarity and congruence among plane figures.

Learning Objectives

After studying this material, students are expected to be able to:

1. distinguish similar and non-similar figures,
2. distinguish congruent and non-congruent figures,
3. determine unknown side lengths in similar figures,
4. state the conditions for congruent triangles,
5. prove that two triangles are similar,
6. apply similarity ratios to solve geometry problems.

A. Similar Plane Figures

1. Two Similar Plane Figures

Consider the following figure.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ZKZ8mIt3mtQ7IcEhkZABSwGgu2UkpldYZYeBiQcblMFXUNzkLYDWy9J5DEcYJvHfEtVufEPRwgPVCKQDBixAWxnS-iRk0Gt7vRX6RUMGLfcJg1Npb6zBINqZf3gJb5hXBNuIK1oQ4Ok/s645/kesebangunan1.png)

Suppose rectangles $ABCD$ and $PQRS$ have corresponding sides:

• $\overline{AB}$ with $\overline{PQ}$
• $\overline{BC}$ with $\overline{QR}$
• $\overline{CD}$ with $\overline{RS}$
• $\overline{AD}$ with $\overline{PS}$

Their side ratios are:

$$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{CD}{RS}=\frac{AD}{PS}=\frac{1}{2}$$

The corresponding angles are also equal:

$$\angle A=\angle P,\quad \angle B=\angle Q,\quad \angle C=\angle R,\quad \angle D=\angle S$$

So:

$$ABCD \simeq PQRS$$

Two plane figures are similar if:

1. corresponding sides are proportional,
2. corresponding angles are equal.

Example

Consider the two kites below.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje57Ka8U_khSvjjx7wqeWJL_dJWzwjCIuDvLXqw82F_MUHDJ2fme2Cw9DMbOoLKHMqTZ2BlI4EidMd6-RdPJ8oi19zKMcbF8SjzOzKGbEYjlIDRi0CBtP25zMCEkTPdM31kekHGSGokyU/s424/kesebangunan2.png)

Using the properties of kites, the corresponding angles are equal and the corresponding sides have the same ratio:

$$\frac{BC}{FG}=\frac{DC}{GH}=\frac{AD}{EH}=\frac{AB}{EF}=\frac{3}{2}$$

Therefore:

$$ABCD \simeq EFGH$$

2. Two Figures with the Same Shape and Size

Consider two identical banknotes.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI1ir1U_jQodqelyRcsIuJMQvycYjp_TN6QGbudoAvnoztN3NyQjYDOqVNMUuzHhglobGeqC8oaVO54v1HyjxelMFc26hK8h7px-WRxJl7sOa_KTCyPY1QtFmSZWA_2YAENi8ookan9aU/s598/kekongruenan.jpg)

If all corresponding sides are equal and all corresponding angles are equal, then the figures are congruent.

Two figures are congruent if:

1. corresponding sides are equal in length,
2. corresponding angles are equal.

Example

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLPU8t53gBSAUo-wNN_dRqS171cf9PLgMBNC6f-XhxAXfMVkScqOfza-aUt3Co5BqMkgJbZ_D0Y_jHk51cdxs0KS_r09fK4ZhIy1Le6UtZ_L9RxHiYKlh6CpIIF6WeYwnOp0ro5uFSPME/s515/kongruen.png)

If all corresponding sides of the two rhombuses have the same length and all corresponding angles are equal, then:

$$ABCD \cong EFGH$$

3. Finding an Unknown Side

Similarity can be used to find an unknown side length.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJQBiMpX0atYxR0E3PN_I2QPKK-cdNB-Mbbjro5XNseyXJom6WbsdEQXQjyNo7oxhkSaR_PP-7gYFCXzoMpp1bG3WguXom50dilJKFoF1MgNfM6bMWu134jO-eqjbER6mjlLdoRJv3TmQ/s474/kesebangunan3.png)

Given:

$$\frac{21}{9}=\frac{7}{3}$$

then:

$$\frac{7}{x}=\frac{7}{3}\Rightarrow x=3$$

and

$$\frac{14}{y}=\frac{7}{3}\Rightarrow y=6$$

So:

$$x=3\text{ cm},\quad y=6\text{ cm}$$

B. Similar Triangles

1. Conditions for Similar Triangles

Consider the figure below.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjirwFZW0aUIfzMvtQs21NnXaXnrwM7UPEKhc8dyAVA0M7fgEozm6B8aX3Urjvd3Q5uT8jZm1ONywf9F2uQQ3jaT9itklYEWsE-Zg9lTBPXYw6xi6_yZ5gb6POedOlynCeeVK4BCtgXnWA/s856/kesebangunan4.png)

If:

$$\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$$

and the corresponding angles are equal, then:

$$\triangle ABC \simeq \triangle DEF$$

In general, two triangles are similar if:

1. corresponding sides are proportional, or
2. two pairs of corresponding angles are equal.

Example

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBQwbXpGi1ce029mlo_k2LmLej3MHyZiCU-Onq_Nc6Q7tAgfhB5gX4ukUVr3hVbr326TnbjQsupqCwcf_-DizYJ3TiJM_rfjnfyQ5lHsuLkLXZz7pvUslXwImQxFC5ERzE88wNpe-tzWI/s446/kongruen1.png)

If:

$$\frac{KL}{PQ}=\frac{LM}{QR}=\frac{KM}{PR}=\frac{2}{3}$$

then:

$$\triangle KLM \simeq \triangle PQR$$

2. Special Similarity in Right Triangles

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRWDMxs_XheXiZ3UMOFiFfI2efQ0TT_u1ay_ji8aveskfxFN3V4-fNRt1IclkcdxORpzYksOv4U95c3Rtd0F4uDVQrD9u_4gIPbIC1uOv3rSAe-uBX0SexWH8M1-tYrKejpOU84bdGfuo/s295/kongruen2.png)

For right triangles, these useful relationships hold:

1. $AD^2=BD\times CD$
2. $AB^2=BD\times BC$
3. $AC^2=CD\times CB$

Example

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAcL4-UHAsf-KJuj9R3FAcyd7wTnIPp2DXRKCvNyy57r1XbGXTj127CwL9FzC0hSOVTjnUzqnuKTE6dDjJiIG4JHVIwId4jTPrhfkcPHhSbo8O_s74wJV9zN27j5rXcSJDE80-ZVRip1U/s287/kongruen3.png)

If $AB=6$ cm and $BC=8$ cm, then:

$$AC=\sqrt{6^2+8^2}=10\text{ cm}$$

Next:

$$AB^2=AD\times AC\Rightarrow 36=10AD\Rightarrow AD=3.6\text{ cm}$$$$DC=10-3.6=6.4\text{ cm}$$$$BD^2=AD\times DC=3.6\times 6.4=23.04$$$$BD=4.8\text{ cm}$$

3. Finding Unknown Sides in Similar Triangles

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq72y5ddBnTZBVnwYLjgT9lZoyh39L88mA-fbOkqgd3lgGLG_pzgHf91fukGAu-5u4-xkgclQMUqKmQBH0Nx20aR_xWjddD-Q8Se4X-ZFXaAlk4iChFrjB2l8WOfsSrtqI4xFdUHRVd5g/s474/kongruen4.png)

If $\triangle ABC \simeq \triangle DEF$, then:

$$\frac{AB}{DE}=\frac{BC}{EF}$$$$\frac{12}{6}=\frac{15}{EF} \Rightarrow EF=7.5\text{ cm}$$

Another example:

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihUjZzch7oIfocLwS9nH6uvd4MGZkSOZdPPFATO2JHcNe87nAcBMwfiXLnhgFUisa7lrYe_YIp3qTD60BUSBA0AKtz_IH6hJTbbD43ni5epjU7ZtAIBJDI768eRYumI_LShxhDImGTe4c/s293/kongruen5.png)

Because $\triangle AED \simeq \triangle BCE$:

$$\frac{AD}{BC}=\frac{AE}{CE}$$$$\frac{9}{16}=\frac{AE}{12} \Rightarrow AE=6.75\text{ cm}$$

4. Parallel Lines on Triangle Sides

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOwyY4A6iCEAKuubHHgifqXjVOdsmyqNAcFzrv_l_USj2QlSQ8pKebR_Bly0vVtCgT93W4auN2owWNcAbRDJWQfbC7c6o5EmEsrz2jeKbdvD1EzPcR_PmF3aXsJHRrIZPxlXHF19nO7LA/s481/kongruen6.png)

If $DE\parallel AB$, then:

$$\triangle DEC \simeq \triangle ABC$$

So:

$$\frac{CD}{AC}=\frac{CE}{BC}=\frac{DE}{AB}$$

This often leads to:

$$\frac{d}{a}=\frac{c}{b}$$

Example

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9lsjToGeqTzQv4CQUmLPG5gG54wIU_RBF3s3vt8quu8Kg-gaQVKKOjEdfRlRK6-6kzgpVsvBhj-FRxrNyUacpmEAd_T6s0sBm9-bUR3ecndZL5MplGR2lcp_56Fz1Bv5n1fxrv8gdJmE/s426/kongruen7.png)

If $\overline{PT}\parallel \overline{QS}$, then:

$$\frac{QS}{PT}=\frac{QR}{PQ+QR}$$$$\frac{3}{4}=\frac{QR}{2+QR} \Rightarrow QR=6\text{ cm}$$

And:

$$\frac{QS}{PT}=\frac{RS}{ST+RS}$$$$\frac{3}{4}=\frac{4}{ST+4} \Rightarrow ST=\frac{4}{3}\text{ cm}$$

5. Word Problems Involving Similarity

Two common applications are:

1. finding the height of a pole using a smaller reference object,
2. finding missing dimensions in a photo and frame setup.

For the pole problem:

$$\frac{2}{t}=\frac{3}{9}\Rightarrow t=6$$

So the height of the pole is 6 m.

For the frame problem:

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGbJAGPh8lQCrQsTcTdJrPk4FAmZJKcDQfBrIB3Hl4jhqAOx0c36AExis7sJQT1tlwZRwhAH5quh4gBPtXrX-pi02SRgUhAROFvaXqFIadRd-peX7ab6EuTWhUolY04D6Gx-re7lebILI/s307/kongruen12.png)

$$\frac{54-x}{60}=\frac{28}{40}=\frac{7}{10}$$$$54-x=42\Rightarrow x=12$$

So the lower frame width is 12 cm.

C. Congruent Triangles

1. Properties of Congruent Triangles

Congruent triangles have:

1. the same shape,
2. the same size,
3. equal corresponding side lengths,
4. equal corresponding angles.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY1KhS3uwpgCLsHzjagznwWF58gG_Dw7QFSMsrsDfj_WGW7iZvWyWussyL6v8nBXfviKaVpWGfDvL7dgXnFZb1i_RkHxSsiHi41QXX02UEAOTIS2f0dDDri5olOCeL0Vs0RI75v7uvReo/s514/kongruen8.png)

2. Conditions for Congruent Triangles

Two triangles are congruent if they satisfy one of these conditions:

1. SSS: all three corresponding sides are equal,
2. SAS: two corresponding sides and the included angle are equal,
3. ASA: two corresponding angles and the included side are equal.

a. SSS

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxfQu3wweo2717qoYWnybjxlCqoAU1jieZP4bA9JsNpI07JEL7-4F1rexBMDVTsufA8KoX5k3uYSTd54YES8VN9eiCuzW740S6rwisPAe3WTOlMT963KqTUbO0xMwtzYb7WKvqOnlLVIs/s480/kongruen9.png)

If:

$$AB=DE,\quad BC=EF,\quad AC=DF$$

then:

$$\triangle ABC \cong \triangle DEF$$

b. SAS

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMSyW8WFjUkzKMd7T86mJudESYC9RrC4B7-fpllf1jDyyYbA49r75KpTDm22Bs6gvrMSLkgqvfovWJlCSKCxi6of0rHYjHFEulXAWcwSVaskj8nV_V3ILhciFM8TS8bIZ8n2n_5RyjD00/s448/kongruen11.png)

If:

$$AB=DE,\quad AC=DF,\quad \angle BAC=\angle EDF$$

then:

$$\triangle ABC \cong \triangle DEF$$

c. ASA

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdRI1oBtJiRc5ROsGFOdUXCbyxMRGQ89OnC2q8fDMU8aEXlpGaenlVPDSqW9uiRdPHnvyun-BJGZelDfypoczD7_dZ940yJzzAUV-SED2DOrl2BNEljwu5pmNtbZQlnNY_l29Mo91yOoU/s425/kongruen10.png)

If:

$$AB=DE,\quad \angle A=\angle D,\quad \angle B=\angle E$$

then:

$$\triangle ABC \cong \triangle DEF$$

Example in a Kite

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLM0hBvOROOw3Ae1pqvEdSlHeSme9n2U5bm_Dduey6pLp1Q7BNYQF_9Rusp1ocPqlqmLwljEqaP88rBB7T-Wx7GqhHWk0XWnG4urc4cE8Swidt6yivAU6akCcBRiWbGuQCD4UaBtc939U/s334/kongruen12.png)

Some congruent pairs that can be identified are:

1. $\triangle AED \cong \triangle ABE$
2. $\triangle DEC \cong \triangle BEC$
3. $\triangle ACD \cong \triangle ABC$

D. Finding Sides and Angles in Congruent Triangles

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoPAegiuVDnz6SJiq1LeKs66CLxI3QW9ZCHWZtJtWZLMIDzSdQWmqsd1yLW_q9LMC2q9juWguxnnY85CTxr082yvR-G7PbqybRmFdMMBeCItjXL-3jBpKBUSAG1aUhyphenhyphenWb_tABxsz3gneE/s310/kongruen13.png)

If $\triangle KNM \cong \triangle LNM$, then the corresponding sides are equal:

$$KM=ML=10\text{ cm},\quad KN=LN=5\text{ cm}$$

The length of $MN$:

$$MN=\sqrt{10^2-5^2}=\sqrt{75}=5\sqrt{3}\text{ cm}$$

If $\angle NKM=60^\circ$, then:

$$\angle MLN=60^\circ$$

and

$$\angle KMN=\angle NML=30^\circ$$

E. GeoGebra Simulation

Move the points in the simulation below to observe how similar and congruent triangles behave.

Notice that:

$$\triangle ABC \sim \triangle DEF$$

and

$$\triangle ABC \cong \triangle A'B'C'$$

F. Evaluation

After studying this material, try the following questions.

1. A person stands 2 meters from a lamppost and casts a 3-meter shadow. If the person’s height is 1.8 meters, what is the height of the lamppost?
2. A photo measuring 12 cm x 15 cm is placed on cardboard. The top, left, and right margins are each 2 cm. If the photo and the cardboard are similar, what is the area of the cardboard?
3. If $AD:DB=2:1$ and $DE=6$ cm, find $BC$.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAfvmUiwjiMsp67EffHapOXv72iMba6cuCCavTUvRCNO2qcHyJ4vM1L0f07BVXwEy68MnawhwO9yfs6TpSbq6o8cgqdZgVRNVk8EUfOEpxZS3XnWwc9VUn6hzdmXpC-tLRwbJPCuMDt7g/s298/kongruen15.png)

4. In a trapezoid with $AB\parallel CD$ and $EF\parallel AB$, if $DE:EA=5:3$, $CD=2$ cm, and $AB=10$ cm, find $EF$.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3yrIjC9oDrg3E2EQtArm8SXf0_4lgWgB0qU_1GQOHZBtPV0cMEb2j_o4wCmO8Ea6_Fgv20eG4aBN0QUjABbXbbX3U5T908T3upTJ8OcbtnFHv8D2XM_UWxQ432pcStFcD8boyWzPdnTg/s353/kongruen16.png)

5. If $AB=CD=8$ cm and $AD=20$ cm, find $BC$.

[

](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38SxRpYzmBStVr5LO46cwnmiS3zQVfXOjDmkiR9-0I1119TzNKqHlw7cqOFsJrYZ9GF_IOAAUgNbmEN1y2mVDYwS6_6jq-ZvRt4ZXttidu5S90PAAinoAQHRvVE7G7Z2dg8BRVKlkpiQ/s317/kongruen17.png)

This material can be used as a foundation for solving many geometry problems involving ratios, proportions, and triangle properties.