Tổng Ba Góc Của Một Tam Giác – Từ Cơ Bản Đến Nâng Cao

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The sum of the three angles of a triangle is an extremely basic knowledge in high school geometry math. So today, Kien Guru would like to share with readers the theories to remember as well as some types of exercises to apply this knowledge. Let’s learn together with Kien Guru:

I. The theory of the sum of the three angles of a triangle.

1. Theorem.

In a triangle, the sum of the measures of the three angles is 180 degrees.

Considering triangle ABC, by theorem we have:

2. Application in right triangle.

Definition: A triangle with one right angle is called a right triangle.

Based on the 7th grade math theorem, the sum of the three angles of a triangle, then in a right triangle, the two acute angles are complementary. Specifically:

3. The property of the exterior angle of the triangle.

Definition: An exterior angle of a triangle is a complementary angle to any other angle in the triangle.

Nature:

– Each exterior angle of a triangle has a measure equal to the sum of two non-adjacent interior angles.– An exterior angle of a triangle has a greater measure than each non-adjacent interior angle.

Specifically, in the triangle ABC below:

Angle ACD is an exterior angle of a triangle.

Based on the above properties, we have:

II. Exercises to apply the sum of the three angles of a triangle.

1. Method.

Based on the relationship between the angles in the triangle:

– The 3 angles in a triangle have a sum of 180 degrees.

– An exterior angle has a measure equal to the sum of two non-adjacent interior angles.- In a right triangle, the two acute angles are complementary.

We will create related equality, from which we will find the required angle.

2. Exercises with solutions.

Exercise 1: Let triangle ABC satisfy:

Calculate angle C value?

Guide:

Considering triangle ABC, we have:

I guess

Exercise 2: Consider an isosceles triangle ABC at A, whose base angle measures 55 degrees. Calculate the measure of the angle at the vertex?

Guide:

Recall knowledge: An isosceles triangle is a triangle with two equal sides, the angle formed by those two sides is the vertex angle, and the other two angles are the base angles. By property, the two base angles are congruent.

Based on the properties of the above isosceles triangle, we have:

Derived from:

Exercise 3: Consider a right triangle ABC at A, angle B has a measure of 40 degrees. Calculate angle B?

Guide:

According to the problem, triangle ABC is right-angled at A, deducing:

So

Exercise 4: Consider an isosceles triangle ABC (AB=AC), the angle at the vertex is 100 degrees. Calculate the measure of the other two angles?

Guide:

Since triangle ABC has AB=AC, it follows that triangle ABC is isosceles at A.

By subject:

.

Based on the property that the two bases of an isosceles triangle are congruent, we have:

Other way:

Derived from:

Exercise 5: Consider triangle ABC satisfying:

. The internal bisector of angle ABD intersects side AC at D. Calculate the values ​​of angles: ADB, angle CDB?

Guide:

Considering triangle ABC, we have:

deduce:

Again, BD is the bisector of angle ABC, so:

Considering triangle BDC whose angle BDA is the exterior angle at vertex D, deduce:

Similarly, considering triangle ABD whose angle BDC is the exterior angle at vertex D, deduce:

So we have the answer we need.

Exercise 6: Let ABC be a triangle with angle A of 100 degrees. Know that:

. Calculate the measure of angle B and angle C?

Guide:

Considering triangle ABC, there are:

By title, we have:

It has the following system:

Exercise 7: Find the x, y values ​​in the following figure:

Guide:

Considering triangle MNP right angled at M, we have:

Similarly we also have:

Exercise 8: Let ABC be a triangle such that AB is perpendicular to AC. Let E be a point inside triangle ABC. Prove that BEC is an obtuse angle.

Guide:

To prove that the angle BEC is obtuse, we can prove it indirectly, that is, prove that the complementary angle to BEC is an acute angle. Specifically, we need to prove:

is an acute angle.

Consider triangle BEC, with angle

is the exterior angle at the vertex E, so:

but:

deduce

is an acute angle.

Again we have:

infer that angle BEC is an obtuse angle.

Lesson 9: Let triangle ABC satisfy

. We draw the bisector of angle A intersecting side BC at point D. Draw line segment AH perpendicular to side BC (H lies on BC). Calculate the measure of angle BAC, angle ADH and angle HAD?

Guide:

Consider triangle ABC with:

deduce:

where AD is the internal bisector of angle BAC, then:

Consider triangle ADC with

is the exterior angle at vertex D, so:

Considering triangle AHD, right angled at H, we have:

Candlestick:

3. Some problems of grade 7 the sum of the three angles of a triangle are self-practice.

Exercise 1: Let ABC be a triangle with AB perpendicular to BC, the measure of angle A is 45 degrees. Calculate angle C? What do you think about this triangle?

Exercise 2: Given a right triangle ABC at A, draw a line AH perpendicular to side BC (H lies on BC).

1. Name the complementary angles.
2. Find pairs of equal acute angles.

Exercise 3: Calculate the value of x in the following figures:

Exercise 4: Draw a triangle ABC with angle A equal to 90 degrees. Know that

.

1. Calculate the remaining angle of the triangle.
2. Draw line segment AH perpendicular to side BC (H belongs to BC). Calculate the measure of angle BAH and angle CAH.

Above is a summary of theory as well as exercises about the sum of three angles of a triangle. Hopefully the article will provide useful knowledge for you, helping you both strengthen and train your math-solving thinking. In addition, you can also refer to other exercises on the Kien Guru app to firmly grasp your knowledge and learn better. Wish you all good study.

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