Geometry 6.5 6.6 Practice Worksheet Answers
Geometry is an essential part of mathematics that deals with the study of shapes, sizes, and positions of objects. It involves understanding various concepts, such as points, lines, angles, and planes, to solve problems and prove theorems. In this blog post, we will provide answers to the Geometry 6.5 6.6 practice worksheet, which covers topics such as congruent triangles, similar triangles, and right triangle trigonometry.
Congruent Triangles (6.5)
Congruent triangles are triangles that have the same size and shape. This means that their corresponding angles and sides are equal. The following theorems can be used to prove that two triangles are congruent:
- Side-Side-Side (SSS) Theorem: If three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent.
- Side-Angle-Side (SAS) Theorem: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
- Angle-Side-Angle (ASA) Theorem: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
Let’s practice with some examples:
Example 1: In the figure below, ∆ABC and ∆DEF are two triangles. If AB = DE = 5 cm, BC = EF = 7 cm, and AC = DF = 3 cm, prove that ∆ABC ≅ ∆DEF.
| Step | Reason |
|---|---|
| 1. AB = DE | Given |
| 2. BC = EF | Given |
| 3. AC = DF | Given |
| 4. ∆ABC ≅ ∆DEF | SSS Theorem |
Example 2: In the figure below, ∆GHI and ∆JKL are two triangles. If GH = JK = 6 cm, ∠G = ∠J = 60°, and HI = KL = 4 cm, prove that ∆GHI ≅ ∆JKL.
| Step | Reason |
|---|---|
| 1. GH = JK | Given |
| 2. ∠G = ∠J | Given |
| 3. HI = KL | Given |
| 4. ∆GHI ≅ ∆JKL | SAS Theorem |
Similar Triangles (6.6)
Similar triangles are triangles that have the same shape but not necessarily the same size. This means that their corresponding angles are equal, and their corresponding sides are proportional.
Let’s practice with some examples:
Example 1: In the figure below, ∆ABC and ∆DEF are two triangles. If AB = 6 cm, BC = 8 cm, AC = 10 cm, DE = 3 cm, EF = 4 cm, and DF = 5 cm, prove that ∆ABC ~ ∆DEF.
| Step | Reason |
|---|---|
| 1. AB/DE = BC/EF | Given |
| 2. AB/DE = AC/DF | Given |
| 3. ∆ABC ~ ∆DEF | AA Similarity Theorem |
Example 2: In the figure below, ∆GHI and ∆JKL are two triangles. If GH = 8 cm, HI = 6 cm, ∠G = 30°, JK = 4 cm, KL = 3 cm, and ∠J = 30°, prove that ∆GHI ~ ∆JKL.
| Step | Reason |
|---|---|
| 1. GH/JK = HI/KL | Given |
| 2. ∠G = ∠J | Given |
| 3. ∆GHI ~ ∆JKL | SAS Similarity Theorem |
📝 Note: In similar triangles, corresponding angles are equal, and corresponding sides are proportional.
Right Triangle Trigonometry
Right triangle trigonometry involves the use of trigonometric ratios to solve problems involving right triangles. The most common trigonometric ratios are:
- Sine (sin): opposite side/hypotenuse
- Cosine (cos): adjacent side/hypotenuse
- Tangent (tan): opposite side/adjacent side
Let’s practice with some examples:
Example 1: In the figure below, ∆ABC is a right triangle. If AB = 3 cm and BC = 4 cm, find the value of sin A.
| Step | Reason |
|---|---|
| 1. sin A = opposite side/hypotenuse | Definition of sine |
| 2. sin A = AB/AC | Substitute values |
| 3. sin A = 3⁄5 | Calculate value |
Example 2: In the figure below, ∆GHI is a right triangle. If GH = 6 cm and HI = 8 cm, find the value of cos G.
| Step | Reason |
|---|---|
| 1. cos G = adjacent side/hypotenuse | Definition of cosine |
| 2. cos G = GH/GI | Substitute values |
| 3. cos G = 6⁄10 | Calculate value |
📝 Note: In right triangle trigonometry, the hypotenuse is always the longest side, opposite the right angle.
In this blog post, we have provided answers to the Geometry 6.5 6.6 practice worksheet, covering topics such as congruent triangles, similar triangles, and right triangle trigonometry. We hope this helps you in your studies!
What is the difference between congruent and similar triangles?
+Congruent triangles have the same size and shape, while similar triangles have the same shape but not necessarily the same size.
What are the three theorems used to prove congruent triangles?
+The three theorems used to prove congruent triangles are the SSS Theorem, SAS Theorem, and ASA Theorem.
What is the definition of sine in right triangle trigonometry?
+The definition of sine in right triangle trigonometry is opposite side/hypotenuse.
