ASA Theorem

We know that triangles can be congruent and also similar to each other. And we always consider all its sides and angles to prove it. But not anymore, here we will learn a triangle criterion by which we can easily prove congruent triangles.

ASA Theorem ASA Theorem

Create learning materials about ASA Theorem with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    In this section, we will take a look at ASA Theorem and understand how to prove congruence and similarity between triangles without all sides and angles of triangles.

    ASA Theorem geometry

    In geometry, two triangles are congruent when either all the sides of one triangle are equal to all sides of another triangle respectively. Or all the three angles of both the triangles should be equal respectively. But with the ASA criterion, we can show congruent triangles with the help of two angles and one side of each triangle.

    ASA theorem, as the name suggests, considers two angles and one side of one triangle equal to another triangle respectively. Here two adjacent angles and the included side between these angles are taken. But one should remember that ASA is not the same as AAS. As ASA has the included side of the two triangles, but in AAS the selected side is the unincluded side of both angles.

    ASA Theorem, ASA triangles, StudySmarterASA triangles, StudySmarter Originals

    ASA similarity and congruence theorem

    We can easily find similar triangles and congruent triangles with the help of the ASA similarity and congruence theorem.

    ASA similarity theorem

    We know that if two triangles are similar then all the corresponding sides are in proportionality and all the corresponding pairs are congruent from the definition of similar triangles. However, in order to ensure the similarity of two triangles we only need information about two angles with the ASA similarity theorem.

    ASA similarity theorem : Two triangles are similar if two corresponding angles of one triangle are congruent to the two corresponding angles of another triangle. Also, the corresponding sides are proportional.

    Mathematically we represent as, if A=X, B=Y, then ABC~XYZ. AndABXY=BCYZ=ACXZ.

    ASA Theorem, ASA similarity triangles, StudySmarterASA Similarity triangles, StudySmarter Originals

    Generally ASA similarity is more well known as the AA similarity theorem, as there is nothing further to check because of only one ratio of sides. Also when two angle measures are given, we can easily find the third angle as the total angle measure isid="2696588" role="math" 180°. So we can easily check the equality of corresponding angles of two triangles and determine the similarity of both triangles.

    ASA congruence theorem

    ASA congruence theorem stands for Angle-Side-Angle and gives the congruent relation between two triangles.

    ASA congruence theorem: Two triangles are congruent if two adjacent angles and the included side on one triangle are congruent to the two angles and included side of another triangle.

    Mathematically we say that, ifBM, BCMN, CN, then id="2696597" role="math" ABCLMN.

    As the angles and sides are congruent they will also be equal. SoB=M, BC=MN, C=N,thenABCLMN.

    ASA Theorem, ASA congruence triangles, StudySmarterASA congruence triangles, StudySmarter Originals

    ASA theorem proof

    Now let us take a look at ASA theorem proof for similarity and congruence.

    ASA similarity theorem proof

    For two triangles ABC and XYZ, it is given from the statement of ASA similarity theorem that A=X, B=Y.

    To prove: ABC~XYZ. And ABXY=BCYZ=ACXZ.

    ASA Theorem, ASA triangles, StudySmarterASA triangle with constructed line , StudySmarter Originals

    Now as two angles A and B are already given in ABC, we can easily find C by taking 180°-(A+B).And the same will be the case for the triangleXYZ.

    We will construct a line PQ in triangle XYZ such that XP=AB and XQ=AC. Also it is given that A=X. Then by using SAS congruence theorem we get that ABC~XPQ.

    Since ABC~XPQ then the corresponding parts of congruent triangles are congruent.

    B=P (1)

    Also, it is given that

    B=Y (2)

    From equation (1) and equation (2) P=Y.

    Since P and Y forms corresponding angles and XY works as transversal PQYZ.

    Using Basic Proportionality Theorem in XYZ,

    XPXY=XQXZ

    Basic Proportionality theorem states that if a line is parallel to one side of a triangle and it intersects the other two sides at two different points then those sides are in proportion.

    From the construction of PQ we replace XP=ABand XQ=AC in the above equation.

    ABXY=ACXZ

    Similarly, ABXY=BCYZ

    ABXY=BCYZ=ACXZ

    Hence ABC~XYZ.

    ASA congruence theorem proof

    We are given from the statement of ASA congruence theorem that B=M, C=N, and BC=MN.

    ASA Theorem, ASA congruence triangles, StudySmarterASA congruent triangles, StudySmarter Originals

    To prove: ABCLMN

    To prove the above statement we will consider different cases.

    Case 1: Assume that AB=LM.

    ASA Theorem, ASA congruence triangles, StudySmarterASA congruent triangles with AB and LM equal, StudySmarter Originals

    In ABC and LMN, AB=LM from our assumption. And it is given that B=M and BC=MN. So by SAS congruence theorem ABCLMN.

    Case 2: Suppose AB>LM.

    Then we construct point X on AB such that id="2696624" role="math" XB=LM.

    ASA Theorem, ASA congruence triangles, StudySmarterASA congruent triangle with constructed point X, StudySmarter Originals

    We have XB=LM, B=M and BC=MN. Using SAS congruence theorem XBCLMN.

    Now it is given that ACB=LNM (1)

    And from the above congruence,id="2696626" role="math" XBCLMN, we get that id="2696647" role="math" XCB=LNM (2)

    So from equations (1) and (2), we get

    ACB=LNM=XCB ACB=XCB

    But from our assumption of AB>LM, and also by looking at the figure this is not possible. So ACB=XCB can only occur when both the points A and X coincides and AC=XC.

    So we are again left with the fact that ABC and XCB are equal. Hence we can consider only one triangle ABCsuch that AB=LM. So this is the same as case 1, and from that we get that ABCLMN.

    Case 3: Suppose AB<LM.

    Then construct a point Y on LM such that AB=YM and we repeat the same argument as in case 2.

    ASA Theorem, ASA congruence triangles, StudySmarterASA congruent triangles with constructed point Y, StudySmarter Originals

    Hence we get that ABCLMN.

    ASA Theorem example

    Let us see some examples related to ASA theorems.

    Calculate BD and CE in the given figure, if ACBD, AE=3 cm, AC=6 cm, BE=4 cm, DE=8 cm.

    Solution:

    In ACE and BDE, as ACBD then A=B because they are alternate interior angles. Also AEC=BED forms vertically opposite angles.

    Then by ASA similarity theorem ACE~BDE.

    We also get from the ASA similarity theorem that CEDE=AEBE=ACBD.

    Then by substituting all the given values in the above equation,

    CE8=34=6BD

    BD=43×6 =8 cm

    And id="2696649" role="math" CE=34×8 =6 cm

    Hence BD=8 cm, CE=6 cm.

    Calculate the value of x when D=55°, F=65°, B=(2x+30)°.

    Solution:

    From the figure we can see that A=D, AC=DF, C=F. Then by ASA congruence theorem we get that ABCDEF.

    A+B+C=180°

    Now substituting all the given values we get,

    55°+(2x+30)°+65°=180°55°+30°+65°+2x=180°150°+2x=180°2x=180°-150°2x=30°x=30°2x=15°

    ASA Theorem - Key takeaways

    • ASA congruence theorem: Two triangles are congruent if two adjacent angles and the included side on one triangle are congruent to the two angles and included side of another triangle.
    • ASA similarity theorem: Two triangles are similar if two corresponding angles of one triangle are congruent to the two corresponding angles of another triangle. Also, the corresponding sides are proportional.
    • ASA similarity is mostly known as the AA similarity theorem.
    • ASA theorem is not the same as the AAS theorem.
    Frequently Asked Questions about ASA Theorem

    What is the ASA theorem?

    Two triangles are congruent if two adjacent angles and the included side on one triangle are congruent to the two angles and included side of another triangle.

    How do you use ASA theorem?

    ASA theorem is used to find congruence between two triangles if two angles and included side are given.

    How do you know if a triangle is ASA?

    If two angles and the included side are given and they are equal to the corresponding angles and side, then that triangles are ASA.

    How do you prove ASA theorem?

    ASA theorem is proved with the help of the SAS theorem.

    Test your knowledge with multiple choice flashcards

    Is ASA theorem same as AAS theorem?

    If two corresponding pairs of angles of two triangles are given and both are congruent, the third pair of angles can be easily found.

    Is the given figure congruent using ASA congruence theorem?(flashcards) en-mathematics-pure maths-geometry-asa triangle common side

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 7 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner