Angle versus Side Length

<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-angle-versus-side-length" target="_blank">Angle versus Side Length Exercise</a></p><p>The <strong><span style="color:#8e44ad;">Relationship between Angle and Side Opposite</span></strong> is pretty straightforward. As the angle gets bigger, the opposite side does as well. As it gets smaller, the opposite side gets smaller too. See below, kindly.</p> <center><svg height="131.22857666015625" style=" width:661.8285522460938px; height:131.22857666015625px; background: #FFF; fill: none; " width="661.8285522460938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="arrow-line"><path class="connection real" d=" M46,112.4 L164.6,72.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M46,112.4 L168.6,111.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M168.6,111.26 L164.6,72.26" stroke-dasharray="6 6" style="stroke: rgb(208, 2, 27); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M182,110.4 L282.6,47.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M182,110.4 L304.6,109.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M304.6,109.26 L282.6,47.26" stroke-dasharray="6 6" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M326,110.4 L396.6,34.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M326,110.4 L448.6,109.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M448.6,109.26 L396.6,34.26" stroke-dasharray="6 6" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M475,111.4 L490.6,25.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M475,111.4 L597.6,110.26" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M597.6,110.26 L490.6,25.26" stroke-dasharray="6 6" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="129.39999389648438" style="width:660px;height:129.39999389648438px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><g transform="matrix(1,0,0,1,94.91424560546875,108.0571533203125)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,209.91424560546875,106.0571533203125)"><path d="M280 181L280 106C280 46 269 32 220 30L158 27L158 -3C291 0 291 0 315 0C339 0 339 0 472 -3L472 27L424 30C375 33 364 46 364 106L364 181C423 181 444 180 472 177L472 248L364 245L365 697L285 667L2 204L2 181ZM280 245L65 245L280 597ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,341.91424560546875,106.0571533203125)"><path d="M131 331C152 512 241 611 421 665L379 689C283 657 242 637 191 593C88 506 32 384 32 247C32 82 112 -20 241 -20C371 -20 468 83 468 219C468 334 399 409 293 409C216 409 184 370 131 331ZM255 349C331 349 382 283 382 184C382 80 331 13 254 13C169 13 123 86 123 220C123 255 127 274 138 291C160 325 207 349 255 349ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,483.914306640625,107.0571533203125)"><path d="M168 345C127 326 110 315 88 294C50 256 30 211 30 159C30 56 113 -20 226 -20C356 -20 464 82 464 206C464 286 427 329 313 381C404 443 436 485 436 545C436 631 365 689 259 689C140 689 53 613 53 508C53 440 80 402 168 345ZM284 295C347 267 385 218 385 164C385 80 322 14 241 14C156 14 101 74 101 167C101 240 130 286 204 331ZM223 423C160 454 128 494 128 544C128 610 176 655 247 655C320 655 368 607 368 534C368 477 343 438 278 396ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g></g></g></svg> </svg></center> <p>Even within a triangle, this relationship holds. A triangle of course consists of three angles. The angle opposite the longest side will be the largest, the angle opposite the second longest side will be the second largest, and the angle opposite the smallest side will be the smallest. See below, if you will.</p> <center><svg height="246.6857147216797" style=" width:661.8285522460938px; height:246.6857147216797px; background: #FFF; fill: none; " width="661.8285522460938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape"><path class="real" d=" M381.6,9.23 L407.6,168.26 L253,168.26 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M299.6,147.26 L388.64,98.84" stroke-dasharray="6 6" style="stroke: rgb(74, 144, 226); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path><g stroke="rgb(74,144,226)" stroke-opacity="1" style="stroke: rgb(74, 144, 226); stroke-width: 1;" transform="matrix(-0.8785671524643952,0.4776188423948579,-0.4776188423948579,-0.8785671524643952,390.39996337890625,97.88571166992188)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g class="arrow-line"><path class="connection real" d=" M371.6,148.26 L324.93,95.75" stroke-dasharray="6 6" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path><g stroke="rgb(208,2,27)" stroke-opacity="1" style="stroke: rgb(208, 2, 27); stroke-width: 1;" transform="matrix(0.6643176677893625,0.7474503570558396,-0.7474503570558396,0.6643176677893625,323.5999755859375,94.25714111328125)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g class="arrow-line"><path class="connection real" d=" M363.6,65.26 L341.07,158.31" stroke-dasharray="6 6" style="stroke: rgb(189, 16, 224); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path><g stroke="rgb(189,16,224)" stroke-opacity="1" style="stroke: rgb(189, 16, 224); stroke-width: 1;" transform="matrix(0.23531174871692045,-0.9719199457341047,0.9719199457341047,0.23531174871692045,340.5999755859375,160.25714111328125)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g></g></g><g></g><g></g><g></g></svg> <svg height="244.85714721679688" style="width:660px;height:244.85714721679688px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><g transform="matrix(1,0,0,1,281.91424560546875,89.93713806152344)"><path d="M418 -3L418 27L366 30C311 33 301 44 301 96L301 700L60 598L67 548L217 614L217 96C217 44 206 33 152 30L96 27L96 -3C250 0 250 0 261 0C292 0 402 -3 418 -3ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,402.91424560546875,95.9371533203125)"><path d="M409 603L47 -1L157 -1L497 659L497 689L44 689L44 477L74 477L81 533C89 595 96 603 142 603Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,332.91424560546875,194.9371533203125)"><path d="M459 253C459 366 378 446 264 446C216 446 180 443 127 396L127 605L432 604L432 689L75 690L75 322L95 316C142 363 169 377 218 377C314 377 374 309 374 201C374 90 310 25 201 25C147 25 97 43 83 69L37 151L13 137C36 80 48 48 62 4C90 -11 130 -20 173 -20C301 -20 459 89 459 253Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,271.91424560546875,160.9371533203125)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414ZM642 689C560 689 493 622 493 539C493 456 560 389 643 389C724 389 793 457 793 537C793 621 726 689 642 689ZM642 649C703 649 753 599 753 537C753 479 703 429 643 429C582 429 533 478 533 539C533 600 582 649 642 649Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,377.91424560546875,159.9371533203125)"><path d="M235 722L223 733C171 707 135 698 63 691L59 670L107 670C131 670 141 663 141 646C141 639 140 628 139 622L38 71C37 68 37 64 37 61C37 22 85 -11 140 -11C177 -11 228 8 271 39C367 107 433 244 433 376C433 414 424 453 412 468C405 477 392 482 377 482C353 482 323 474 295 460C244 433 211 403 149 324ZM322 424C348 424 361 401 361 352C361 288 340 202 310 137C277 68 237 36 183 36C137 36 112 59 112 101C112 135 127 276 208 361C241 395 293 424 322 424ZM661 689C579 689 512 622 512 539C512 456 579 389 662 389C743 389 812 457 812 537C812 621 745 689 661 689ZM661 649C722 649 772 599 772 537C772 479 722 429 662 429C601 429 552 478 552 539C552 600 601 649 661 649Z" fill="rgb(208,2,27)" fill-opacity="1" stroke="rgb(208,2,27)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,361.91424560546875,58.93713806152344)"><path d="M342 330L365 330C373 395 380 432 389 458C365 473 330 482 293 482C248 483 175 463 118 400C64 352 25 241 25 136C25 40 67 -11 147 -11C201 -11 249 9 304 54L354 95L346 115L331 105C259 57 221 40 186 40C130 40 101 80 101 159C101 267 136 371 185 409C206 425 230 433 261 433C306 433 342 414 342 390ZM605 689C523 689 456 622 456 539C456 456 523 389 606 389C687 389 756 457 756 537C756 621 689 689 605 689ZM605 649C666 649 716 599 716 537C716 479 666 429 606 429C545 429 496 478 496 539C496 600 545 649 605 649Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,291.91424560546875,223.9371533203125)"><path d="M235 722L223 733C171 707 135 698 63 691L59 670L107 670C131 670 141 663 141 646C141 639 140 628 139 622L38 71C37 68 37 64 37 61C37 22 85 -11 140 -11C177 -11 228 8 271 39C367 107 433 244 433 376C433 414 424 453 412 468C405 477 392 482 377 482C353 482 323 474 295 460C244 433 211 403 149 324ZM322 424C348 424 361 401 361 352C361 288 340 202 310 137C277 68 237 36 183 36C137 36 112 59 112 101C112 135 127 276 208 361C241 395 293 424 322 424Z" fill="rgb(208,2,27)" fill-opacity="1" stroke="rgb(208,2,27)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,310.5714111328125,223.9371533203125)"><path d="M65 0L604 252L604 273L65 528L65 469L503 264L65 59Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,334.27142333984375,223.9371533203125)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,352.46429443359375,223.9371533203125)"><path d="M65 0L604 252L604 273L65 528L65 469L503 264L65 59Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,376.16424560546875,223.9371533203125)"><path d="M342 330L365 330C373 395 380 432 389 458C365 473 330 482 293 482C248 483 175 463 118 400C64 352 25 241 25 136C25 40 67 -11 147 -11C201 -11 249 9 304 54L354 95L346 115L331 105C259 57 221 40 186 40C130 40 101 80 101 159C101 267 136 371 185 409C206 425 230 433 261 433C306 433 342 414 342 390Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g></svg> </svg></center> <p><strong><span style="color:#e74c3c;">WARNING</span>:&nbsp;</strong>This relationship is NOT proportional. For example, notice that the side opposite angle $b$ is $10$ and the side opposite angle $c$ is $5$. Does this mean that angle $b$ is twice as large as $c$? Nope! The relationship doesn&#39;t work like that. They&#39;re not that close.&nbsp;</p>