AA Criterion of Similarity

Here we will prove the theorems related to AA Criterion of Similarity on Quadrilateral.

1. In a right-angled triangle, if a perpendicular is drawn from the right-angled vertex to the hypotenuse, the triangles on each side of it are similar to the whole triangle and to one another.

Solution:

Given: Let XYZ be a right angle in which ∠YXZ = 90° and XM ⊥ YZ.

AA Criterion of Similarity

Therefore, ∠XMY = ∠XMZ = 90°.

To prove: ∆XYM ∼ ∆ZXM ∼ ∆ ZYX.

Proof:

            Statement

            Reason

1. In ∆XYM and ∆XYZ,

(i) ∠XMY = ∠YXZ = 90°.

(ii) ∠XYM = ∠XMZ

1.

(i) Given.

(ii) Common angle.

2. Therefore, ∆XYM ∼ ∆ZYX.

2. By AA criterion of similarity.

3. In ∆XYZ and ∆XMZ,

(i) ∠YXZ = ∠XMZ = 90°.

(ii) ) ∠XZY= ∠XZM.

3.

(i) Given.

(ii) Common angle.

4. Therefore, ∆ZYX ∼ ∆ ZXM.

4. By AA criterion of similarity.

5. Therefore, ∆XYM ∼ ∆ZXM ∼ ∆ ZYX. (Proved)

5. From statement 2 and 4.


2. If in the ∆XYZ, ∠X = 90° and XM ⊥ YZ, M being the foot of the perpendicular, prove that XM\(^{2}\) = YM ∙ MZ.

AA Criterion of Similarity Problem

Solution:

In ∆XMY and ∆ZMX,

∠XMY = ∠ZMX = 90°

∠YXM = ∠XZM, because ∠XYM + ∠YXM = 90° = ∠XZM + ∠XYM

⟹ ∠YXM = ∠XZM

Therefore, ∆XMY ∼ ∆ZMX, (by AA criterion of similarity)

Therefore, \(\frac{XM}{ZM}\) = \(\frac{YM}{XM}\)

⟹ XM\(^{2}\) = YM ∙ MZ. (Proved)


3. In the two similar triangles PQR and XYZ, PM ⊥ QR and XN ⊥ YZ. Prove that \(\frac{PQ}{XY}\) = \(\frac{PM}{XN}\).

Solution:

Proof:

            Statement

            Reason

1. In ∆PQM and ∆XYN,

(i) ∠PQM = ∠XYN

(ii) ∠PMQ = ∠XNY = 90°

1.

(i) Being similar triangles, they are equiangular.

(ii) Given

2. ∆PQM ∼ ∆XYN

2. By AA criterion of similarity.

3. \(\frac{PQ}{XY}\) = \(\frac{PM}{XN}\). (Proved)

3. Corresponding sides of similar triangles are proportional.








9th Grade Math

From AA Criterion of Similarity to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

RSS

  1. Worksheet on Yesterday, Today and Tomorrow | Sequence of the Week-days

    Nov 25, 24 01:18 AM

    Practice the worksheet on yesterday, today and tomorrow, the questions are based on the sequence of the week-days, their names and order. We know, the present day is today, the day before today

    Read More

  2. Yesterday, Today and Tomorrow | Name of each Day of the Week

    Nov 25, 24 01:09 AM

    We will learn the relation between yesterday, today and tomorrow. There are seven days in a week. We know the name of each day of the week. Sunday is known as the first day of the week.

    Read More

  3. The Story about Seasons | Spring | Summer | Autumn | Winter

    Nov 25, 24 12:48 AM

    The Four Seasons
    Kids let’s enjoy the story about seasons. Here we will discuss about the four seasons and the duration. Some months are too hot and some are too cold. The period of hot months is called the hot

    Read More

  4. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Nov 25, 24 12:17 AM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  5. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Nov 24, 24 11:01 PM

    Time Duration Example
    Time duration tells us how long it takes for an activity to complete. We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton…

    Read More