Solved Exercises on Triangles and Their Properties in Grade 7

Master triangles and their properties: classification, angle sum, exterior angles, and triangle inequality through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Triangle Classification
Exercise 1
Classify the triangle with side lengths 5 cm, 5 cm, and 8 cm. Also classify it by its angles.
Definition:

Triangle Classification by Sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (all sides different).

Method for triangle classification:
  1. Compare all three side lengths
  2. Determine if any sides are equal
  3. Classify by sides (equilateral, isosceles, scalene)
  4. Determine angle type by comparing sides
Given
5, 5, 8 cm
By Sides
Isosceles
By Angles
Acute
Step 1: Compare side lengths

Two sides are equal (5 cm and 5 cm)

One side is different (8 cm)

Step 2: Classify by sides

Since two sides are equal, it's an isosceles triangle

Step 3: Classify by angles

Use the relationship: if c² < a² + b², then the triangle is acute

Here, 8² = 64 and 5² + 5² = 25 + 25 = 50

Since 64 > 50, the triangle is obtuse

Step 4: Verify with angle sum

Using the law of cosines to find the largest angle:

cos(C) = (a² + b² - c²)/(2ab) = (25 + 25 - 64)/(2×5×5) = -14/50 = -0.28

Since cos(C) is negative, angle C is obtuse (>90°)

Isosceles and obtuse triangle
Final answer:

The triangle is isosceles (by sides) and obtuse (by angles).

Applied rules:

• Side Classification: Two equal sides = isosceles

• Angle Classification: c² > a² + b² → obtuse triangle

• Triangle Types: Equilateral, isosceles, scalene by sides

2 Angle Sum Property
Exercise 2
Two angles of a triangle measure 40° and 70°. Find the measure of the third angle and classify the triangle by its angles.
Definition:

Triangle Angle Sum Property: The sum of all interior angles in any triangle is always 180°.

Given
40°, 70°
Formula
A + B + C = 180°
Third Angle
70°
Step 1: Apply the angle sum property

∠A + ∠B + ∠C = 180°

Step 2: Substitute known values

40° + 70° + ∠C = 180°

Step 3: Solve for the third angle

110° + ∠C = 180°

∠C = 180° - 110° = 70°

Step 4: Classify by angles

All angles are less than 90° (40°, 70°, 70°)

Therefore, it's an acute triangle

Step 5: Classify by sides

Two angles are equal (70° each), so two sides are equal

Therefore, it's also an isosceles triangle

Third angle = 70°, Triangle is acute and isosceles
Final answer:

The third angle measures 70°. The triangle is acute and isosceles.

Applied rules:

• Angle Sum Property: Interior angles sum to 180°

• Angle Classification: All angles < 90° = acute

• Equal Angles: Equal angles correspond to equal sides

3 Triangle Inequality
Exercise 3
Determine if triangles can be formed with the following sets of side lengths: (a) 3 cm, 4 cm, 5 cm; (b) 2 cm, 7 cm, 10 cm.
Definition:

Triangle Inequality Theorem: The sum of any two sides of a triangle must be greater than the third side.

Case (a)
3, 4, 5
Case (b)
2, 7, 10
Results
(a) Yes, (b) No
Step 1: State the triangle inequality

For sides a, b, c: a + b > c, a + c > b, b + c > a

Step 2: Test case (a) with sides 3, 4, 5

3 + 4 = 7 > 5 ✓

3 + 5 = 8 > 4 ✓

4 + 5 = 9 > 3 ✓

All inequalities satisfied, so a triangle can be formed

Step 3: Test case (b) with sides 2, 7, 10

2 + 7 = 9 < 10 ✗

2 + 10 = 12 > 7 ✓

7 + 10 = 17 > 2 ✓

One inequality fails, so a triangle cannot be formed

Step 4: Conclusion

Case (a): Triangle possible

Case (b): Triangle not possible

Step 5: Additional insight for case (a)

Since 3² + 4² = 9 + 16 = 25 = 5², this is a right triangle (Pythagorean triple)

(a) Yes, (b) No
Final answer:

(a) A triangle can be formed with sides 3 cm, 4 cm, 5 cm.

(b) A triangle cannot be formed with sides 2 cm, 7 cm, 10 cm.

Applied rules:

• Triangle Inequality: Sum of any two sides > third side

• Verification: Check all three combinations

• Pythagorean Triple: 3-4-5 is a special right triangle

Triangle Properties and Theorems
\(\angle A + \angle B + \angle C = 180°\)
Triangle Angle Sum
By Sides
Equilateral, Isosceles, Scalene
Classification based on side lengths
By Angles
Acute, Right, Obtuse
Classification based on angle measures
Inequality
a + b > c
Triangle formation condition
Key definitions:

Triangle: A polygon with three sides and three angles.

Equilateral Triangle: A triangle with all three sides equal and all angles equal to 60°.

Isosceles Triangle: A triangle with at least two sides equal and the base angles equal.

Scalene Triangle: A triangle with all sides of different lengths.

Acute Triangle: A triangle where all angles are less than 90°.

Right Triangle: A triangle with one angle equal to 90°.

Obtuse Triangle: A triangle with one angle greater than 90°.

Complete methodology:
  1. Identify Given Information: Note side lengths, angle measures, or other properties
  2. Choose Classification Method: Determine if classifying by sides or angles
  3. Apply Relevant Theorem: Use angle sum property, triangle inequality, etc.
  4. Verify Conditions: Check that all requirements are met
  5. Draw Conclusion: State the classification with justification
  6. Check Reasonableness: Ensure the answer makes geometric sense
Tip 1: Remember: all triangles have angles that sum to 180°.
Tip 2: The longest side is always opposite the largest angle.
Tip 3: For triangle inequality, only one failing condition means no triangle.
Tip 4: Equal angles correspond to equal sides in a triangle.
Common errors: Forgetting the 180° sum, misapplying triangle inequality, confusing side and angle classifications.
Exam preparation: Practice with all classification types, work with algebraic expressions for angles, memorize key theorems.
Formulas to know by heart:

• Angle Sum: ∠A + ∠B + ∠C = 180°

• Triangle Inequality: a + b > c, a + c > b, b + c > a

• Exterior Angle: Exterior angle = sum of two remote interior angles

• Pythagorean Theorem: a² + b² = c² (for right triangles)

• Area: (1/2) × base × height

• Perimeter: a + b + c

Solution: Exercises 4 to 5
4 Exterior Angle Theorem
Exercise 4
In triangle ABC, ∠A = 50° and ∠B = 60°. Find the measure of the exterior angle at vertex C.
Definition:

Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.

Given
∠A = 50°, ∠B = 60°
Theorem
Ext∠C = ∠A + ∠B
Result
110°
Step 1: Apply the exterior angle theorem

Exterior angle at C = ∠A + ∠B

Step 2: Substitute known values

Exterior angle at C = 50° + 60° = 110°

Step 3: Verify with linear pair

Interior ∠C = 180° - 50° - 60° = 70°

Exterior ∠C + Interior ∠C = 180° (linear pair)

110° + 70° = 180° ✓

Step 4: Alternative verification

Using the fact that exterior angle = 180° - interior angle

Exterior ∠C = 180° - 70° = 110° ✓

Exterior angle at C = 110°
Final answer:

The measure of the exterior angle at vertex C is 110°.

Applied rules:

• Exterior Angle Theorem: Ext∠ = sum of remote interior angles

• Linear Pair: Interior + exterior = 180°

• Angle Sum: Interior angles sum to 180°

5 Multiple Properties Problem
Exercise 5
A triangle has angles in the ratio 2:3:4. Find the measures of all three angles and classify the triangle by its angles. Then determine if the sides could be in the same ratio.
Definition:

Angle Ratios: When angles are in a given ratio, we can express them as multiples of a variable and use the angle sum property to solve.

Ratio
2:3:4
Angles
40°, 60°, 80°
Classification
Acute triangle
Step 1: Set up the ratio

Let the angles be 2x, 3x, and 4x degrees

Step 2: Apply the angle sum property

2x + 3x + 4x = 180°

9x = 180°

x = 20°

Step 3: Find all angles

First angle: 2x = 2(20°) = 40°

Second angle: 3x = 3(20°) = 60°

Third angle: 4x = 4(20°) = 80°

Step 4: Classify by angles

All angles are less than 90° (40°, 60°, 80°)

Therefore, it's an acute triangle

Step 5: Consider if sides could be in same ratio

For sides to be in ratio 2:3:4, check triangle inequality:

2 + 3 = 5 > 4 ✓

2 + 4 = 6 > 3 ✓

3 + 4 = 7 > 2 ✓

Yes, a triangle with sides in ratio 2:3:4 is possible

Step 6: Note the relationship

Important: The ratio of sides is NOT the same as the ratio of angles

Side ratios depend on the sine of the angles (Law of Sines)

Angles: 40°, 60°, 80°; Classification: Acute; Side ratio possible
Final answer:

The angles measure 40°, 60°, and 80°. The triangle is acute. Yes, sides could be in the ratio 2:3:4.

Applied rules:

• Ratio Representation: Express as multiples of a variable

• Angle Sum Property: Sum to 180°

• Triangle Inequality: Check possibility of side ratios

Complete Guide: Triangles and Their Properties
\(\text{If } c^2 = a^2 + b^2 \text{, then triangle is right}\)
Pythagorean Relationship
Key definitions:

Triangle: A three-sided polygon formed by connecting three non-collinear points. It has three vertices, three sides, and three interior angles.

Vertices: The corner points of the triangle (usually labeled A, B, C).

Sides: The line segments connecting the vertices (usually labeled a, b, c, opposite to angles A, B, C respectively).

Interior Angles: The angles inside the triangle formed by the intersection of two sides.

Exterior Angles: The angles formed when one side of the triangle is extended.

Complete methodology:
  1. Identify Triangle Type: Determine if you need to classify by sides or angles
  2. Apply Relevant Properties: Use angle sum, inequality, or other theorems
  3. Set Up Equations: Represent relationships algebraically when needed
  4. Solve Systematically: Use algebra to find unknown values
  5. Verify Solutions: Check that all triangle properties are satisfied
  6. Draw Conclusions: State the final classification or answer
Tip 1: The largest angle is always opposite the longest side.
Tip 2: Equal sides correspond to equal angles in any triangle.
Tip 3: The exterior angle is always greater than either of the remote interior angles.
Tip 4: Remember: 180° is the magic number for interior angles!
Common errors: Forgetting the 180° sum, misapplying the triangle inequality, confusing angle and side relationships.
Exam preparation: Practice all classification types, memorize the key theorems, work with algebraic expressions for angles and sides.
Formulas to know by heart:

• Angle Sum: ∠A + ∠B + ∠C = 180°

• Triangle Inequality: a + b > c, a + c > b, b + c > a

• Exterior Angle: Ext∠ = sum of two remote interior angles

• Pythagorean Theorem: a² + b² = c² (right triangles only)

• Area: (1/2) × base × height

• Perimeter: a + b + c

• Angle-Side Relationship: Largest angle ↔ longest side

• Equal Sides ↔ Equal Angles

Visualizing Triangle Types: Classification by Angles
Exercise 6: Triangle Angle Classification
Consider different triangle types based on angle measures:
Acute (<90° each), Right (=90° one), Obtuse (>90° one)
Showing how angles determine triangle classification

Analysis: The chart shows how different angle ranges classify triangles.

  • Acute triangles have all angles less than 90°
  • Right triangles have exactly one 90° angle
  • Obtuse triangles have exactly one angle greater than 90°
  • The sum of all angles is always 180°

Questions & Answers

Question: Can a triangle have two right angles or two obtuse angles? Why or why not?

Answer: No, a triangle cannot have two right angles or two obtuse angles. Here's why:

For two right angles:

  • If two angles were 90° each, their sum would be 180°
  • The third angle would be 180° - 180° = 0°
  • A 0° angle cannot form a triangle

For two obtuse angles:

  • If two angles were obtuse (each > 90°), their sum would be > 180°
  • This would make the third angle negative, which is impossible

Therefore, a triangle can have at most one right angle or one obtuse angle, with the other two angles being acute.

Question: Is there a relationship between the sides and angles of a triangle? If one side is longer, is its opposite angle always larger?

Answer: Yes, there is a direct relationship between sides and angles in a triangle:

Angle-Side Relationship:

  • The largest angle is always opposite the longest side
  • The smallest angle is always opposite the shortest side
  • If two sides are equal, their opposite angles are equal (and vice versa)

Examples:

  • In a triangle with sides 3, 4, 5: the 5-side is opposite the largest angle
  • In an isosceles triangle: the equal sides are opposite equal angles
  • In an equilateral triangle: all sides and angles are equal (60° each)

This relationship is consistent in all triangles and is a fundamental property of triangle geometry.

Question: How can I remember the different types of triangles? It's hard to keep them all straight.

Answer: Here are some memory aids for triangle types:

By Sides:

  • Equilateral: "Equal" - all sides are equal (think "equator" for equal)
  • Isosceles: "I saw" - I saw two equal sides (isosceles has two 's' like two equal sides)
  • Scalene: "Scale" - all sides are on a different scale (different lengths)

By Angles:

  • Acute: "A cute" - all angles are "a cute" little angle (<90°)
  • Right: "Right" angle = 90° (just right!)
  • Obtuse: "Obvious" - the obtuse angle is obviously large (>90°)

You can also remember that triangles can be classified by both sides AND angles simultaneously (like "acute isosceles" or "right scalene").