{"id":120,"date":"2022-04-13T08:23:59","date_gmt":"2022-04-13T08:23:59","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=120"},"modified":"2025-10-24T06:17:57","modified_gmt":"2025-10-24T06:17:57","slug":"g-10-congruency-in-isosceles-and-equilateral-triangles","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/g-10-congruency-in-isosceles-and-equilateral-triangles\/","title":{"rendered":"G.10 Congruency in isosceles and equilateral triangles"},"content":{"rendered":"\n

Congruency in isosceles and equilateral triangles<\/strong><\/h2>\n\n\n\n

key notes :<\/strong><\/p>\n\n\n\n

\ud83d\udd3a What is Congruency?<\/strong><\/h3>\n\n\n\n

\ud83d\udc49 Two figures are said to be congruent<\/strong> if they have the same shape and size<\/strong>.
\ud83d\udff0 In triangles, congruency means all corresponding sides and angles are equal.<\/strong><\/p>\n\n\n\n

\n\n\n\n

\ud83d\udc6f\u200d\u2642\ufe0f Isosceles Triangle<\/strong><\/h3>\n\n\n\n

\ud83d\udcd8 Definition:<\/strong> A triangle with two sides of equal length<\/strong> and the angles opposite those sides are also equal.<\/strong>
\ud83d\udca1 Example:<\/strong> If AB=AC, then \u2220B=\u2220C.<\/p>\n\n\n\n

\n\n\n\n

\ud83d\udca0 Congruency in Isosceles Triangles<\/strong><\/h3>\n\n\n\n

\u2b50 If two sides<\/strong> of a triangle are equal,
then the angles opposite those sides are also equal.<\/strong>
\ud83d\udccf Similarly, if two angles<\/strong> are equal,
then the sides opposite those angles<\/strong> are equal.<\/p>\n\n\n\n

\ud83e\udde9 Hence, the two base angles are congruent!<\/strong><\/p>\n\n\n\n

\n\n\n\n

\ud83d\udd39 Congruency Criteria for Isosceles Triangles<\/strong><\/h3>\n\n\n\n

\ud83e\uddee You can use any of the triangle congruence rules<\/strong>:
\u2705 SSS (Side-Side-Side)<\/strong>
\u2705 SAS (Side-Angle-Side)<\/strong>
\u2705 ASA (Angle-Side-Angle)<\/strong>
\u2705 AAS (Angle-Angle-Side)<\/strong>
\u2705 RHS (Right angle-Hypotenuse-Side)<\/strong><\/p>\n\n\n\n

\n\n\n\n

\ud83d\udd37 Equilateral Triangle<\/strong><\/h3>\n\n\n\n

\ud83d\udcd8 Definition:<\/strong> A triangle with all sides equal<\/strong> and all angles equal (each 60\u00b0).<\/strong>
\ud83d\udff0 Every equilateral triangle is also isosceles<\/strong> (because it has at least two equal sides).<\/p>\n\n\n\n

\n\n\n\n

\ud83d\udcab Congruency in Equilateral Triangles<\/strong><\/h3>\n\n\n\n

\u2b50 Any two equilateral triangles are congruent<\/strong> if one side of one<\/strong> equals one side of the other.<\/strong>
\ud83e\udde0 Reason: All sides and all angles are equal in an equilateral triangle!<\/p>\n\n\n\n

\n\n\n\n

\ud83e\udded Important Properties Summary<\/strong><\/h3>\n\n\n\n
\ud83d\udd39 Property<\/th>\ud83d\udd38 Isosceles Triangle<\/th>\ud83d\udd38 Equilateral Triangle<\/th><\/tr><\/thead>
Equal sides<\/td>2 sides<\/td>3 sides<\/td><\/tr>
Equal angles<\/td>2 angles<\/td>3 angles (each 60\u00b0)<\/td><\/tr>
Line of symmetry<\/td>1<\/td>3<\/td><\/tr>
Congruency condition<\/td>Base angles or sides equal<\/td>One side equal is enough<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

\ud83d\udc96 Real-Life Examples<\/strong><\/h3>\n\n\n\n

\ud83c\udfe0 Roofs of houses (isosceles triangles)
\ud83c\udf88 Triangular road signs (equilateral triangles)
\ud83c\udfaa Decorative patterns, flags, and logos<\/p>\n\n\n\n

\n\n\n\n

\ud83c\udfaf Key Takeaway<\/strong><\/h3>\n\n\n\n

\ud83d\udc49 All equilateral triangles are isosceles, but not all isosceles triangles are equilateral!<\/strong>
\u2728 Congruency helps us prove equality<\/strong> in sides and angles of these triangles.<\/p>\n\n\n\n

Learn with an example<\/p>\n\n\n\n

\n
\n

What is the value of p?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

p =<\/p>\n<\/div><\/div>\n\n\n\n

Look at the diagram.<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

ST and SU are marked with one hatch mark each. So, they are congruent. By the Isosceles Triangle Theorem, the angles opposite ST and SU must also be congruent. <\/p>\n\n\n\n

The angle opposite ST is \u2220U and the angle opposite SU is \u2220T. So, \u2220U and \u2220T have the same measure. From the diagram you can see that \u2220T=p, so \u2220U=p as well.<\/p>\n\n\n\n

Now, set the sum of the interior angle measures of \u25b3STU equal to 180\u00b0 and solve for p.<\/p>\n\n\n\n

\u2220S+\u2220T+\u2220U = 180\u00b0
48\u00b0+p+p = 180\u00b0 —-> Plug in \u2220S=48\u00b0, \u2220T=p and \u2220U=p
2p+48\u00b0 = 180\u00b0 —–> Combine like terms
2p = 132\u00b0 ——>Subtract 48\u00b0 from both sides
p= 66\u00b0 ——->Divide both sides by 2<\/p>\n\n\n\n

So, p=66\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

What is the value of a?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

a= ______\u00b0<\/p>\n<\/div><\/div>\n\n\n\n

Look at the diagram.<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

WX and XY are marked with one hatch mark each. So, they are congruent.<\/p>\n\n\n\n

By the Isosceles Triangle Theorem, the angles oppositeWX and XY must also be congruent.<\/p>\n\n\n\n

The angle opposite WX is \u2220Y and the angle opposite XY is \u2220W. So, \u2220Y and \u2220W have the same measure. From the diagram you can see that \u2220Y=a, so \u2220W=a as well.<\/p>\n\n\n\n

Now, set the sum of the interior angle measures of \u25b3WXY equal to 180\u00b0 and solve for a.<\/p>\n\n\n\n

\u2220W+\u2220X+\u2220Y = 180\u00b0<\/p>\n\n\n\n

a+46\u00b0+a = 180\u00b0 Plug in \u2220W=a, \u2220X=46\u00b0 and \u2220Y=a<\/p>\n\n\n\n

2a+46\u00b0 = 180\u00b0 Combine like terms<\/p>\n\n\n\n

2a = 134\u00b0 Subtract 46\u00b0 from both sides<\/p>\n\n\n\n

a = 67\u00b0 Divide both sides by 2<\/p>\n\n\n\n

So, a=67\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

What is the value of w?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

w=<\/p>\n<\/div><\/div>\n\n\n\n

Look at the diagram.<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

\u2220C and \u2220E are marked with one arc each. So, they are congruent.<\/p>\n\n\n\n

By the Isosceles Triangle Theorem, the sides opposite \u2220C and \u2220E must also be congruent.<\/p>\n\n\n\n

The side opposite \u2220C is DE and the side opposite \u2220E is CD . So, DE and CD have the same length. From the diagram you can see that the length of CD is 27 and the length of DE is w.<\/p>\n\n\n\n

For these to be the same, w must equal 27.<\/p>\n<\/div><\/div>\n\n\n\n

Let’s try some problems!\u270d\ufe0f<\/strong><\/p>\n\n\n\n

\n
\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n\n
\n
\"\"<\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Congruency in isosceles and equilateral triangles key notes : \ud83d\udd3a What is Congruency? \ud83d\udc49 Two figures are said to be congruent if they have the same shape and size.\ud83d\udff0 In triangles, congruency means all corresponding sides and angles are equal. \ud83d\udc6f\u200d\u2642\ufe0f Isosceles Triangle \ud83d\udcd8 Definition: A triangle with two sides of equal length and theContinue reading “G.10 Congruency in isosceles and equilateral triangles”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-120","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=120"}],"version-history":[{"count":31,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/120\/revisions"}],"predecessor-version":[{"id":18129,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/120\/revisions\/18129"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}